WEBVTT 00:00:01.680 --> 00:00:04.080 The following content is provided under a Creative 00:00:04.080 --> 00:00:05.620 Commons license. 00:00:05.620 --> 00:00:07.920 Your support will help MIT OpenCourseWare 00:00:07.920 --> 00:00:12.280 continue to offer high-quality educational resources for free. 00:00:12.280 --> 00:00:14.910 To make a donation or view additional materials 00:00:14.910 --> 00:00:18.870 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.870 --> 00:00:19.770 at ocw.mit.edu. 00:00:22.590 --> 00:00:24.210 HAIM SOMPOLINSKY: My topic today is 00:00:24.210 --> 00:00:26.170 discussing sensory representations 00:00:26.170 --> 00:00:29.950 in deep cortex-like architectures. 00:00:29.950 --> 00:00:33.430 I should say the topic is perhaps 00:00:33.430 --> 00:00:38.300 toward a theory of sensory representations 00:00:38.300 --> 00:00:39.930 in deep networks. 00:00:39.930 --> 00:00:45.540 As you will see, our attempt is to develop 00:00:45.540 --> 00:00:49.680 a systematic theoretical understanding of the capacity 00:00:49.680 --> 00:00:54.420 and limitations of architectures of that type. 00:00:54.420 --> 00:00:58.530 The general context is well known. 00:00:58.530 --> 00:01:02.160 In many sensory systems, we see information 00:01:02.160 --> 00:01:07.770 is propagating from the periphery, like the retina, 00:01:07.770 --> 00:01:12.060 to primary visual cortex, and then, of course, many stages up 00:01:12.060 --> 00:01:17.250 to a very high level, or maybe the hippocampal structure. 00:01:17.250 --> 00:01:18.450 It's not purely feedforward. 00:01:18.450 --> 00:01:23.130 There are backward, massive backward or top-down 00:01:23.130 --> 00:01:25.200 connections, the recurrent connections, 00:01:25.200 --> 00:01:28.410 and some of those extra features I'll talk about. 00:01:28.410 --> 00:01:32.790 But the most intuitive feature of that 00:01:32.790 --> 00:01:36.000 is simply a transformation or filtering 00:01:36.000 --> 00:01:38.910 of data across multiple stages. 00:01:38.910 --> 00:01:42.540 Similarly in auditory pathway. 00:01:42.540 --> 00:01:46.380 In other systems, we see a similar structure or aspect 00:01:46.380 --> 00:01:49.042 of similar structure as well. 00:01:49.042 --> 00:01:53.280 A well-known and a classical formative system 00:01:53.280 --> 00:01:56.550 for computational science is cerebellum, 00:01:56.550 --> 00:02:00.300 where you have information coming from the mossy fiber 00:02:00.300 --> 00:02:04.920 layer, then expand enormously into a granule layer, 00:02:04.920 --> 00:02:08.590 and then converge to a Purkinje cell. 00:02:08.590 --> 00:02:10.889 So if you look at a single Purkinje 00:02:10.889 --> 00:02:14.190 cell, the output of the cerebellum, as a unit, 00:02:14.190 --> 00:02:18.030 then you see there is, first, an expansion 00:02:18.030 --> 00:02:21.570 from 1,000 to two orders of magnitude 00:02:21.570 --> 00:02:23.760 higher in the granule layer. 00:02:23.760 --> 00:02:29.330 And then convergence of 200,000 or so parallel fibers 00:02:29.330 --> 00:02:31.350 onto a single Purkinje cell. 00:02:31.350 --> 00:02:34.350 And there are, those type of modules, many, many 00:02:34.350 --> 00:02:36.250 across the cerebellum. 00:02:36.250 --> 00:02:40.170 So, again, a transformation which involves, in this case, 00:02:40.170 --> 00:02:43.710 expansion and then convergence. 00:02:43.710 --> 00:02:47.490 In the basal ganglia, which I wouldn't categorize it 00:02:47.490 --> 00:02:50.194 as a sensory system. 00:02:50.194 --> 00:02:51.110 More related to motor. 00:02:51.110 --> 00:02:54.540 Nevertheless, you see cortex converging first 00:02:54.540 --> 00:02:57.300 to various stages of the basal ganglia, 00:02:57.300 --> 00:03:01.680 and then expanding again to cortex. 00:03:01.680 --> 00:03:05.220 Hippocampus has also multiple pathways, but some of them 00:03:05.220 --> 00:03:07.380 include a convergence. 00:03:07.380 --> 00:03:13.810 For instance, convergence to a CA3, and then expansion 00:03:13.810 --> 00:03:16.110 again to cortex. 00:03:16.110 --> 00:03:19.680 But there are other multiple pathways as well, 00:03:19.680 --> 00:03:23.790 different stages of sensory information propagating, 00:03:23.790 --> 00:03:27.600 of course, across them. 00:03:27.600 --> 00:03:30.900 And, finally, the artificial network story 00:03:30.900 --> 00:03:34.816 of deep neural networks, all of you may have heard. 00:03:34.816 --> 00:03:38.780 Input layer, then sequence of stages. 00:03:38.780 --> 00:03:40.440 Purely feedforward. 00:03:40.440 --> 00:03:46.800 And at least the canonical leading network 00:03:46.800 --> 00:03:51.720 is one that the output layer has object recognition, object 00:03:51.720 --> 00:03:52.990 classification task. 00:03:52.990 --> 00:03:56.860 And the whole network is studied by backprop, 00:03:56.860 --> 00:04:01.100 supervised learning for that. 00:04:01.100 --> 00:04:07.140 What I'll talk about is more in the spirit of the idea 00:04:07.140 --> 00:04:10.540 that the first stages are more general purpose 00:04:10.540 --> 00:04:16.060 than the specific classic task in the output layer. 00:04:16.060 --> 00:04:17.550 So there are many issues. 00:04:17.550 --> 00:04:20.630 Number of stages that are required, the size of them, 00:04:20.630 --> 00:04:23.520 why compression or expansion. 00:04:23.520 --> 00:04:27.120 In many systems, you'll see that the fraction of active neurons 00:04:27.120 --> 00:04:31.330 is small in the expanded layer. 00:04:31.330 --> 00:04:32.660 That's what we call sparseness. 00:04:32.660 --> 00:04:35.010 So high sparseness means small number 00:04:35.010 --> 00:04:38.340 of neurons active at any given stimulus. 00:04:38.340 --> 00:04:42.180 It's just the terminology is somewhat confusing. 00:04:42.180 --> 00:04:47.110 So high sparseness is small number of active neurons. 00:04:47.110 --> 00:04:51.990 One important and crucial question is how to transform. 00:04:51.990 --> 00:04:53.910 What are the filters, the weights 00:04:53.910 --> 00:04:58.200 that are good for transforming sensory information from one 00:04:58.200 --> 00:04:59.560 layer to another? 00:04:59.560 --> 00:05:04.440 And, in particular, whether random weights 00:05:04.440 --> 00:05:10.380 is good enough, or maybe even optimal in some sense. 00:05:10.380 --> 00:05:13.345 Or one needs more structure, the more 00:05:13.345 --> 00:05:16.080 learned type of synaptic way. 00:05:16.080 --> 00:05:19.620 This is a crucial question, perhaps not for machine 00:05:19.620 --> 00:05:22.920 learning but for computational neuroscience, because there 00:05:22.920 --> 00:05:25.500 is some experimental evidence, for at least 00:05:25.500 --> 00:05:29.560 of some of the systems that are studied, 00:05:29.560 --> 00:05:36.180 that the mapping from the compressed, original 00:05:36.180 --> 00:05:39.600 representation to the sparse representation 00:05:39.600 --> 00:05:43.440 is actually done by randomly connected weights. 00:05:43.440 --> 00:05:46.560 So one example is olfactory cortex. 00:05:46.560 --> 00:05:49.710 The mapping of olfactory representation 00:05:49.710 --> 00:05:53.040 from the olfactory bulb, so from glomerulus layer, 00:05:53.040 --> 00:05:57.486 to the piriform cortex seems to be random, 00:05:57.486 --> 00:05:59.370 as far as one can say. 00:05:59.370 --> 00:06:01.710 Similarly, in the cerebellum, the example 00:06:01.710 --> 00:06:04.200 that I mentioned before, when one 00:06:04.200 --> 00:06:06.840 looks at the mapping from the mossy fiber 00:06:06.840 --> 00:06:10.500 to the granule cell, again enormous expansion 00:06:10.500 --> 00:06:12.000 by a few orders of magnitude. 00:06:12.000 --> 00:06:14.580 Nevertheless, they seem to be random. 00:06:14.580 --> 00:06:18.370 Now, of course, one cannot say exclusively that they are 00:06:18.370 --> 00:06:21.690 random and there are no subtle correlations or structures. 00:06:21.690 --> 00:06:24.780 But, nevertheless, there is a strong motivation 00:06:24.780 --> 00:06:29.620 to ask whether random projections are good enough. 00:06:29.620 --> 00:06:33.330 And if not, what does it mean structured? 00:06:33.330 --> 00:06:38.820 What kind of structure is appropriate for this task? 00:06:38.820 --> 00:06:40.950 Question of top-down and feedback loops, 00:06:40.950 --> 00:06:43.660 recurrent connections, and so on. 00:06:43.660 --> 00:06:48.720 So that's all I hope to, at least briefly, mention 00:06:48.720 --> 00:06:50.430 later on in my talk. 00:06:50.430 --> 00:06:56.640 So before I continue, most of, or a large part of the talk 00:06:56.640 --> 00:07:00.660 will be based on published and unpublished work 00:07:00.660 --> 00:07:03.050 with Baktash Babadi, who was, until recently, 00:07:03.050 --> 00:07:07.320 a postdoctoral, a Swartz Fellow at Harvard University. 00:07:07.320 --> 00:07:10.370 Went to practice medicine. 00:07:10.370 --> 00:07:13.990 Elia Frankin, a master student at the Hebrew University. 00:07:13.990 --> 00:07:18.440 SueYeon, who all you know here, at Harvard. 00:07:18.440 --> 00:07:22.060 Uri Cohen, a PhD student at the Hebrew University. 00:07:22.060 --> 00:07:25.740 And Dan Lee from Penn University. 00:07:25.740 --> 00:07:29.160 So here is our formalization of the problem. 00:07:29.160 --> 00:07:34.350 We have an input layer, denoted 0. 00:07:34.350 --> 00:07:39.750 Typically, it's a small, it's a compressed layer 00:07:39.750 --> 00:07:41.350 with dense representation. 00:07:41.350 --> 00:07:45.240 So here, every input will generate maybe half 00:07:45.240 --> 00:07:48.990 of the population, on average, 00:07:48.990 --> 00:07:51.600 Then there is a feedforward layer 00:07:51.600 --> 00:07:55.140 of synaptic weights, which expand 00:07:55.140 --> 00:08:00.530 to a higher-dimension layer, which we call cortical layer. 00:08:00.530 --> 00:08:03.460 It's expanded in terms of the number of neurons. 00:08:03.460 --> 00:08:05.760 So this will be S1. 00:08:05.760 --> 00:08:09.600 It is sparse because the f, the fraction of neurons 00:08:09.600 --> 00:08:15.790 that are active for each given input vector, will be small. 00:08:15.790 --> 00:08:19.110 So it is expanded and sparse. 00:08:19.110 --> 00:08:22.710 That will be the first part of my talk, discussing this. 00:08:22.710 --> 00:08:28.830 Then, later on, I'll talk about staging, cascading 00:08:28.830 --> 00:08:32.400 this transformation to several stages. 00:08:32.400 --> 00:08:35.190 And ultimately there is a readout, 00:08:35.190 --> 00:08:37.750 will be some classification task. 00:08:37.750 --> 00:08:40.210 1 will be one classification. 00:08:40.210 --> 00:08:43.500 2 will be another classification rule, et cetera, 00:08:43.500 --> 00:08:45.460 each one of them with synaptic weights 00:08:45.460 --> 00:08:49.330 which are learned to perform that task. 00:08:49.330 --> 00:08:51.960 So we call that a supervised layer, and that's 00:08:51.960 --> 00:08:54.300 the unsupervised layer. 00:08:54.300 --> 00:08:56.710 So that's a formalization of the problem. 00:08:56.710 --> 00:09:01.230 And, as you will see, we'll make enormously simplifying 00:09:01.230 --> 00:09:05.490 abstraction of the real biological system 00:09:05.490 --> 00:09:08.010 in order to try to gain some insight 00:09:08.010 --> 00:09:12.240 about the computational capacity of such systems. 00:09:12.240 --> 00:09:14.640 So the first important question is 00:09:14.640 --> 00:09:18.540 what is the statistics, the statistical structure 00:09:18.540 --> 00:09:19.360 of the input? 00:09:19.360 --> 00:09:21.570 So the input is kind of n-dimensional vector, 00:09:21.570 --> 00:09:25.360 where n, or n0, when n is the number of units here. 00:09:25.360 --> 00:09:30.430 So each sensory event evokes a pattern of activity here. 00:09:30.430 --> 00:09:34.440 But what is the structure, the statistical structure, 00:09:34.440 --> 00:09:36.750 that we are working with? 00:09:36.750 --> 00:09:41.390 And the simplest one that we are going to discuss 00:09:41.390 --> 00:09:42.160 is the following. 00:09:42.160 --> 00:09:44.580 So we assume, basically, that the inputs 00:09:44.580 --> 00:09:47.700 are coming from a mixture of, so to speak, 00:09:47.700 --> 00:09:51.450 a mixture of Gaussian statistics. 00:09:51.450 --> 00:09:54.120 It's not going to be Gaussian because, for simplicity, we'll 00:09:54.120 --> 00:09:55.590 assume they are binary. 00:09:55.590 --> 00:09:58.320 But this doesn't matter actually. 00:09:58.320 --> 00:10:04.850 So imagine that this is kind of a graphical represent-- 00:10:04.850 --> 00:10:07.940 a caricature of high-dimensional space. 00:10:07.940 --> 00:10:11.690 And imagine the inputs, the sensory inputs, 00:10:11.690 --> 00:10:16.375 imagine that they are clustered around templates or cluster 00:10:16.375 --> 00:10:16.875 centers. 00:10:19.920 --> 00:10:24.590 So these will be the centers of these balls. 00:10:24.590 --> 00:10:30.080 And the input itself is coming from the neighborhoods 00:10:30.080 --> 00:10:31.590 of those templates. 00:10:31.590 --> 00:10:35.870 So each input will be one point in this space, 00:10:35.870 --> 00:10:41.090 and it will be originating from one of those ensembles of one 00:10:41.090 --> 00:10:42.720 of those states. 00:10:42.720 --> 00:10:46.280 So that's a simple architecture. 00:10:46.280 --> 00:10:47.930 And we are going-- 00:10:47.930 --> 00:10:52.850 in real space, it will be mapping 00:10:52.850 --> 00:10:56.780 from one of those states into another state 00:10:56.780 --> 00:10:59.030 in the next layer. 00:10:59.030 --> 00:11:03.590 And then, finally, the task will be to take-- 00:11:03.590 --> 00:11:07.370 imagine that some of those balls are classified as plus. 00:11:07.370 --> 00:11:12.140 Let's say the olfactory factory stimuli, and some of them 00:11:12.140 --> 00:11:16.630 are classified as appetitive, some of them as aversive. 00:11:16.630 --> 00:11:21.335 So the output layer, the readout unit, 00:11:21.335 --> 00:11:26.360 has to classify some of those spheres as a plus, 00:11:26.360 --> 00:11:28.100 and some of them are minus. 00:11:28.100 --> 00:11:30.680 And, of course, depending on how many of 00:11:30.680 --> 00:11:33.850 them are in a dimensionality, and their location, 00:11:33.850 --> 00:11:36.560 this may or may not be an easy problem. 00:11:36.560 --> 00:11:41.125 So, for instance, here it's fine. 00:11:41.125 --> 00:11:45.740 There is-- a linear classifier on the input space can do it. 00:11:45.740 --> 00:11:50.850 Here, I think there are, there should be some mistakes here. 00:11:50.850 --> 00:11:52.250 Yeah, here. 00:11:52.250 --> 00:11:59.150 So here is a case where the linear classifier at the input 00:11:59.150 --> 00:12:00.020 layer cannot do it. 00:12:00.020 --> 00:12:06.350 And that's our-- that's a theme which 00:12:06.350 --> 00:12:12.890 is very popular, both in computation neural science 00:12:12.890 --> 00:12:19.070 and system neural science studies in machine learning. 00:12:19.070 --> 00:12:22.280 And the following question comes up. 00:12:22.280 --> 00:12:25.310 Suppose we see that there is a transformation of data 00:12:25.310 --> 00:12:34.790 from, let's say, a photoreceptor layer in vision to the ganglion 00:12:34.790 --> 00:12:36.950 cells at the output of the retina, 00:12:36.950 --> 00:12:40.040 then to cortex in several stages. 00:12:40.040 --> 00:12:47.111 How do we gauge, how do we assess 00:12:47.111 --> 00:12:52.610 what is the advantage for the brain to transform information 00:12:52.610 --> 00:12:57.600 from one, let's say, from retina to V1, and so on and so forth. 00:12:57.600 --> 00:13:01.640 After all, in this feedforward's architecture, 00:13:01.640 --> 00:13:06.660 no net information is generated at the next layer. 00:13:06.660 --> 00:13:09.950 So if no net information is generated, 00:13:09.950 --> 00:13:15.090 the question is, what did we gain by these transformations? 00:13:15.090 --> 00:13:22.520 And one possible answer is that it is reformatted, reformatting 00:13:22.520 --> 00:13:26.480 the sensory representation into different representation which 00:13:26.480 --> 00:13:30.020 will make subsequent computations simpler. 00:13:30.020 --> 00:13:34.520 So what does it mean, subsequent computation is simpler? 00:13:34.520 --> 00:13:40.130 One notion of simplicity is whether subsequent computation 00:13:40.130 --> 00:13:45.320 can be realized by a simple linear readout. 00:13:45.320 --> 00:13:53.180 So that's the strategy that we are going to adopt here. 00:13:53.180 --> 00:13:56.510 And this is to ask, as the information, 00:13:56.510 --> 00:13:58.760 as the representation is changing 00:13:58.760 --> 00:14:01.130 as you go from one layer to another, 00:14:01.130 --> 00:14:06.890 how well a linear readout will be able to perform the task. 00:14:06.890 --> 00:14:08.710 So that's the input. 00:14:08.710 --> 00:14:10.710 That's the story. 00:14:10.710 --> 00:14:13.760 And then, as I said, there is an input, 00:14:13.760 --> 00:14:16.526 unsupervised representations, and supervised at the end. 00:14:19.670 --> 00:14:21.960 I need to introduce notations. 00:14:21.960 --> 00:14:22.560 Bear with me. 00:14:22.560 --> 00:14:24.680 This is a computational talk. 00:14:24.680 --> 00:14:28.580 I cannot just talk about ideas, because the whole thing is 00:14:28.580 --> 00:14:33.710 to be able to actually come up with a quantitative theory that 00:14:33.710 --> 00:14:35.150 tests ideas. 00:14:35.150 --> 00:14:38.150 So let me introduce notations. 00:14:38.150 --> 00:14:41.780 So at the centers, at each layer, 00:14:41.780 --> 00:14:45.020 you can ask what is the representation of the centers 00:14:45.020 --> 00:14:47.000 of these stimuli? 00:14:47.000 --> 00:14:50.060 And I'll denote the center by a bar. 00:14:50.060 --> 00:14:53.050 And mu is index of the patterns. 00:14:53.050 --> 00:14:56.900 So mu goes from 1 to P. P Is the number of those balls, 00:14:56.900 --> 00:14:59.870 number of those spheres, or number of those clusters, 00:14:59.870 --> 00:15:03.580 if you think about clustering some sensory data. 00:15:03.580 --> 00:15:05.920 So P would be the number of clusters. 00:15:05.920 --> 00:15:11.622 i, from 1 to N, is simply the neuron or the unit activation 00:15:11.622 --> 00:15:13.720 at each mu. 00:15:13.720 --> 00:15:14.830 And L is the layer. 00:15:14.830 --> 00:15:16.270 So 0 is the input layer. 00:15:16.270 --> 00:15:19.680 It's up to L layer. 00:15:19.680 --> 00:15:21.060 So this would be 0, 1. 00:15:21.060 --> 00:15:29.670 The mean activation at each layer from 1 00:15:29.670 --> 00:15:31.890 on will just have to be a constant 00:15:31.890 --> 00:15:34.500 to be f. f goes from 0 to 1. 00:15:34.500 --> 00:15:38.040 The smaller f is, the sparser the representation is. 00:15:38.040 --> 00:15:41.800 We will assume that the input representation is dense. 00:15:41.800 --> 00:15:43.900 So this is 0.5. 00:15:43.900 --> 00:15:46.860 N, again, we'll assume to be, for simplicity, 00:15:46.860 --> 00:15:49.820 a constant across layers, except for the first layer, where 00:15:49.820 --> 00:15:51.270 there is expansion. 00:15:51.270 --> 00:15:55.320 You can vary those parameters, and actually the theory 00:15:55.320 --> 00:15:58.210 accommodates variations of those. 00:15:58.210 --> 00:15:59.880 But that's the simplest architecture. 00:15:59.880 --> 00:16:02.730 You expend a dense representation 00:16:02.730 --> 00:16:05.460 into a sparse higher dimension, and you 00:16:05.460 --> 00:16:09.810 keep doing it as you go along. 00:16:09.810 --> 00:16:12.460 So that's notation. 00:16:12.460 --> 00:16:21.950 Now, how do we assess what is the next stages doing 00:16:21.950 --> 00:16:23.370 to those clusters. 00:16:23.370 --> 00:16:28.080 So, as I said, one measure is take a linear classifier 00:16:28.080 --> 00:16:31.380 and see how linear classifier performs. 00:16:31.380 --> 00:16:36.720 But, actually, you can also look at the statistics 00:16:36.720 --> 00:16:42.330 of the injected sensory stimuli at each layer 00:16:42.330 --> 00:16:44.250 and learn something from it. 00:16:44.250 --> 00:16:47.250 And, basically, I'm going to suggest 00:16:47.250 --> 00:16:51.450 looking at two major statistical aspects 00:16:51.450 --> 00:16:55.470 of the data in each layer of the transformation. 00:16:55.470 --> 00:16:59.310 One of them is noise, and one of them is correlation. 00:16:59.310 --> 00:17:01.080 So what is noise? 00:17:01.080 --> 00:17:06.960 So, again, noise will be simply the radius, or measure 00:17:06.960 --> 00:17:10.619 of the radius of the sphere. 00:17:10.619 --> 00:17:15.540 So if you had only the templates as inputs, 00:17:15.540 --> 00:17:16.859 the problem would be simple. 00:17:16.859 --> 00:17:20.260 Problem would be easy as long as we have enough dimension. 00:17:20.260 --> 00:17:21.390 You expand it. 00:17:21.390 --> 00:17:24.510 You can easily do linear classifier 00:17:24.510 --> 00:17:25.800 and solve the problem. 00:17:25.800 --> 00:17:28.800 So the problem, in our case, is the fact 00:17:28.800 --> 00:17:31.590 that the input is actually the infinite number of inputs, 00:17:31.590 --> 00:17:34.110 or exponentially large number of possible inputs, 00:17:34.110 --> 00:17:42.870 because they all come from a Gaussian or a binarized version 00:17:42.870 --> 00:17:44.960 of a Gaussian noise around the templates. 00:17:44.960 --> 00:17:47.700 And I'll denote the noise by delta. 00:17:47.700 --> 00:17:51.240 0 means no noise. 00:17:51.240 --> 00:17:54.930 The normalization is such that 1 means that they are random. 00:17:54.930 --> 00:18:00.360 So delta equals to 1 means that, basically, you cannot tell, 00:18:00.360 --> 00:18:03.540 the input, whether it's coming from here or from any other 00:18:03.540 --> 00:18:05.920 points in the input space. 00:18:05.920 --> 00:18:10.980 The other thing, correlations, is more subtle. 00:18:10.980 --> 00:18:14.820 So I'm going to assume that those balls are 00:18:14.820 --> 00:18:17.730 coming from kind of uniform distribution. 00:18:17.730 --> 00:18:20.097 Imagine you take a template here. 00:18:20.097 --> 00:18:21.180 You draw a ball around it. 00:18:21.180 --> 00:18:22.013 You take a template. 00:18:22.013 --> 00:18:23.080 Here, you draw a ball. 00:18:23.080 --> 00:18:26.400 Everything is kind of uniformly distributed. 00:18:26.400 --> 00:18:28.590 The only structure is the fact that data 00:18:28.590 --> 00:18:31.440 comes from this mixture of Gaussians 00:18:31.440 --> 00:18:38.940 or noisy patterns around those centers. 00:18:38.940 --> 00:18:39.840 So that's fine. 00:18:39.840 --> 00:18:48.150 But as you project those clusters into the next stage, 00:18:48.150 --> 00:18:52.530 I claim that those centers, those templates, 00:18:52.530 --> 00:18:56.205 get new representation, which can actually have structure 00:18:56.205 --> 00:19:01.260 in them, simply by the fact that you put all of them 00:19:01.260 --> 00:19:06.400 into this common synaptic weights into the next layer. 00:19:06.400 --> 00:19:11.850 And I'm going to measure this by Q. And, basically, low Q or 0 00:19:11.850 --> 00:19:15.550 Q is basically a kind of randomly uniformly distributed 00:19:15.550 --> 00:19:16.650 centers. 00:19:16.650 --> 00:19:20.100 And I'll always start from that at the input layer. 00:19:20.100 --> 00:19:23.490 But then there is a danger, or it 00:19:23.490 --> 00:19:26.370 might happen that, as you propagate 00:19:26.370 --> 00:19:28.940 this information or this representation 00:19:28.940 --> 00:19:32.725 through the next layer, the centers will look like that, 00:19:32.725 --> 00:19:35.350 or the data, structure of the data, looks like that. 00:19:35.350 --> 00:19:39.450 So, on average, the distance between two centers, 00:19:39.450 --> 00:19:41.340 on average, is the same as here. 00:19:41.340 --> 00:19:43.080 But they are clumped together. 00:19:43.080 --> 00:19:47.010 It's kind of random clustering of the clusters. 00:19:47.010 --> 00:19:51.150 And that can be induced by the fact 00:19:51.150 --> 00:19:58.080 that the data is feedforwarded from this representation. 00:19:58.080 --> 00:20:00.660 That can pose a problem. 00:20:00.660 --> 00:20:03.550 If there is no noise, then there is, again, no problem. 00:20:03.550 --> 00:20:06.500 You can differentiate between them, and so on. 00:20:06.500 --> 00:20:09.250 But if there is noise, this can aggravate the situation, 00:20:09.250 --> 00:20:14.410 because some of the clusters become dangerously close 00:20:14.410 --> 00:20:16.060 to each other. 00:20:16.060 --> 00:20:17.140 And we will come to it. 00:20:17.140 --> 00:20:19.750 But, anyway, so we have this delta, the noise, 00:20:19.750 --> 00:20:23.800 the size of the clusters, and we have Q, the correlations, 00:20:23.800 --> 00:20:28.280 how they are clumped in each representation. 00:20:28.280 --> 00:20:30.970 And now we can ask how delta evolve 00:20:30.970 --> 00:20:32.500 when you go from one presentation 00:20:32.500 --> 00:20:36.340 to another, how Q evolve from one presentation to another, 00:20:36.340 --> 00:20:38.800 and how linear classifier performance will 00:20:38.800 --> 00:20:41.510 change from one representation to another. 00:20:41.510 --> 00:20:48.820 So the simplicity of this assumption 00:20:48.820 --> 00:20:56.540 allows for a kind of systematic, analytical exploration or study 00:20:56.540 --> 00:20:57.040 of this. 00:21:00.400 --> 00:21:02.290 These are definitions. 00:21:02.290 --> 00:21:03.340 Let's go on. 00:21:03.340 --> 00:21:06.490 So what will be the ideal situation? 00:21:06.490 --> 00:21:08.560 So the ideal situation will be that I 00:21:08.560 --> 00:21:14.971 start from some level of noise, which is 00:21:14.971 --> 00:21:18.100 my spheres at the input layer. 00:21:18.100 --> 00:21:20.890 I may or may not start with some correlation. 00:21:20.890 --> 00:21:25.120 The simplest case would be that I start from randomly 00:21:25.120 --> 00:21:26.480 distributed centers. 00:21:26.480 --> 00:21:28.270 So this would be 0. 00:21:28.270 --> 00:21:30.920 And the best situation will be that, 00:21:30.920 --> 00:21:35.920 as I propagate the sensory stimuli, delta, the noise, 00:21:35.920 --> 00:21:38.560 will go to 0. 00:21:38.560 --> 00:21:40.180 As I said, if the noise goes to 0, 00:21:40.180 --> 00:21:42.580 you are left with basically points. 00:21:42.580 --> 00:21:45.670 And those points, if there is enough dimensionality, 00:21:45.670 --> 00:21:48.470 those points would be easily classifiable. 00:21:48.470 --> 00:21:51.580 It would also be good, if the noise 00:21:51.580 --> 00:21:57.110 doesn't go to 0, to have also kind of uniformly spread 00:21:57.110 --> 00:21:57.660 clusters. 00:21:57.660 --> 00:22:00.340 So it will be good to keep Q to be small. 00:22:04.630 --> 00:22:06.514 So let's look at one layer. 00:22:06.514 --> 00:22:07.430 So let's look at this. 00:22:07.430 --> 00:22:10.730 We have the input layer, the output layer here, 00:22:10.730 --> 00:22:13.690 and the readout. 00:22:13.690 --> 00:22:19.340 The first question is what to choose for this input layer. 00:22:19.340 --> 00:22:23.170 So the simplest answer would be choose random. 00:22:23.170 --> 00:22:27.410 So what we do, we just take Gaussian. 00:22:27.410 --> 00:22:31.660 The Gaussian weights in this layer are very simple. 00:22:31.660 --> 00:22:33.140 0 mean, with some normalization. 00:22:33.140 --> 00:22:34.990 It doesn't matter. 00:22:34.990 --> 00:22:39.130 Then we project them into each one of these guys here. 00:22:39.130 --> 00:22:44.971 And then we add threshold to enforce the sparsity 00:22:44.971 --> 00:22:46.120 that we want. 00:22:46.120 --> 00:22:54.790 So whatever the activation here is, whatever the input here is, 00:22:54.790 --> 00:22:57.130 the threshold makes sure that only 00:22:57.130 --> 00:23:01.900 the f with the largest input will be active, 00:23:01.900 --> 00:23:03.347 and the rest will be 0. 00:23:03.347 --> 00:23:05.680 So there is a nonlinearity, which is of course extremely 00:23:05.680 --> 00:23:06.280 important. 00:23:06.280 --> 00:23:11.350 If you map one layer to another with a linear transformation, 00:23:11.350 --> 00:23:15.320 you don't gain anything in terms of classification. 00:23:15.320 --> 00:23:20.560 So there is a nonlinearity, simply a threshold nonlinearity 00:23:20.560 --> 00:23:24.730 after an input projection. 00:23:24.730 --> 00:23:25.360 All right. 00:23:25.360 --> 00:23:27.460 So how we are going to do this? 00:23:27.460 --> 00:23:31.270 So it's straightforward to actually compute analytically 00:23:31.270 --> 00:23:32.890 what will happen to a noise. 00:23:32.890 --> 00:23:36.820 So imagine you take two input vectors with some Hamming 00:23:36.820 --> 00:23:38.860 distance apart from each other. 00:23:38.860 --> 00:23:46.120 You map them by convolving them, so to speak, 00:23:46.120 --> 00:23:49.030 with Gaussian weights, and then thresholding 00:23:49.030 --> 00:23:50.620 them to get some sparsity. 00:23:50.620 --> 00:23:52.420 So f is the sparsity. 00:23:52.420 --> 00:23:55.120 The smaller the f is, the sparser it is. 00:23:55.120 --> 00:23:59.835 So this is the noise level, the normalized Gaussian-- 00:23:59.835 --> 00:24:04.570 I'm sorry-- sphere radius or Hamming distance 00:24:04.570 --> 00:24:07.950 in the output layer, and also the input layer. 00:24:07.950 --> 00:24:10.870 Well if 0 at start, then of course you start at the origin. 00:24:10.870 --> 00:24:13.300 If you are random at the input, you will be random there. 00:24:13.300 --> 00:24:14.790 So these points are fine. 00:24:14.790 --> 00:24:16.960 But, as you see, immediately there 00:24:16.960 --> 00:24:22.210 is an amplification of the noise as you go from the input 00:24:22.210 --> 00:24:23.660 to the output. 00:24:23.660 --> 00:24:29.080 So you start from 0.2, but you get actually, after one layer, 00:24:29.080 --> 00:24:31.480 to 0.6. 00:24:31.480 --> 00:24:36.610 And, actually, the sparser it is-- 00:24:36.610 --> 00:24:39.820 so this is a relatively high sparsity, 00:24:39.820 --> 00:24:43.690 or at least you go from here to here by increasing sparsity, 00:24:43.690 --> 00:24:45.670 namely f becomes smaller. 00:24:45.670 --> 00:24:48.010 And as f becomes smaller, this curve 00:24:48.010 --> 00:24:50.150 is actually steeper and steeper. 00:24:50.150 --> 00:24:54.460 So not only you amplify noise, but you 00:24:54.460 --> 00:24:58.245 also-- the amplification becomes worse the sparser 00:24:58.245 --> 00:25:00.510 the representation is. 00:25:00.510 --> 00:25:06.710 So that is the kind of negative result. 00:25:06.710 --> 00:25:13.300 The idea that you can gain by expanding data to a higher 00:25:13.300 --> 00:25:17.290 dimension and make them more separable 00:25:17.290 --> 00:25:23.290 later on dates back to David Marr's classical theory 00:25:23.290 --> 00:25:24.940 of the cerebellum. 00:25:24.940 --> 00:25:28.090 But what we show here is that, if you think not about 00:25:28.090 --> 00:25:31.510 clean data, a set of points that you want to separate, 00:25:31.510 --> 00:25:34.795 but you think about the more realistic case of you 00:25:34.795 --> 00:25:40.362 have noisy data, or data with high variance, 00:25:40.362 --> 00:25:43.400 then the situation is very different. 00:25:43.400 --> 00:25:48.130 So a random expansion actually amplifies those. 00:25:48.130 --> 00:25:50.200 And that's a theme that will-- 00:25:50.200 --> 00:25:53.170 actually, we will live with it as we go along. 00:25:53.170 --> 00:25:58.910 Random expansion is doing the separation of the templates. 00:25:58.910 --> 00:26:01.720 But the problem is it also separates two nearby points 00:26:01.720 --> 00:26:03.140 within a cluster. 00:26:03.140 --> 00:26:04.480 It also separates them. 00:26:04.480 --> 00:26:08.480 So everything becomes separated from each other. 00:26:08.480 --> 00:26:12.580 And this is why noise is amplified. 00:26:12.580 --> 00:26:16.090 Now, what about the most subtle thing, which is the kind of 00:26:16.090 --> 00:26:19.065 overlap between the centers? 00:26:19.065 --> 00:26:23.780 So, on average, the centers are as far apart as random things. 00:26:23.780 --> 00:26:25.480 But if you look, not on average, but you 00:26:25.480 --> 00:26:28.630 look at the individual pairs, you 00:26:28.630 --> 00:26:30.680 see that there is an excess correlations 00:26:30.680 --> 00:26:32.690 or overlap between them. 00:26:32.690 --> 00:26:34.600 So this is overlap between the centers. 00:26:34.600 --> 00:26:37.900 Again, on average, it is 0, but the variance is not 0. 00:26:37.900 --> 00:26:39.820 On average it's like random, but the variance 00:26:39.820 --> 00:26:43.330 is different, is larger than random. 00:26:43.330 --> 00:26:46.730 And there is an amplification. 00:26:46.730 --> 00:26:51.250 There is a generation of this excess overlap, 00:26:51.250 --> 00:26:53.710 although it's nicely controlled by sparsity. 00:26:53.710 --> 00:26:59.080 So as sparsity goes down, these correlations go down. 00:26:59.080 --> 00:27:02.160 So that's not a tremendous problem. 00:27:02.160 --> 00:27:05.990 The major problem, as I said, is the noise. 00:27:05.990 --> 00:27:12.740 By the way, you can nicely do an exercise where you generate, 00:27:12.740 --> 00:27:17.930 you look at this cortical layer representation, 00:27:17.930 --> 00:27:21.140 and you do SVM, or PCA, and you look 00:27:21.140 --> 00:27:22.950 at the eigenvalue spectrum. 00:27:22.950 --> 00:27:27.710 So if you just look at random sparse points, and you look 00:27:27.710 --> 00:27:32.360 at the SVD this is the eigenvalues number ranked-- 00:27:32.360 --> 00:27:33.690 then that's what you find. 00:27:33.690 --> 00:27:39.020 It's the famous Marchenko-Pastur distribution. 00:27:39.020 --> 00:27:43.800 But, in our case, you see there is an extra power. 00:27:43.800 --> 00:27:46.220 In this case, the input layer is 100, 00:27:46.220 --> 00:27:49.010 so the extra power in the input layer, 00:27:49.010 --> 00:27:51.890 in the first input eigenvalue. 00:27:51.890 --> 00:27:54.640 Now, why is it so? 00:27:54.640 --> 00:27:57.490 What Q is telling us, what nonzero Q 00:27:57.490 --> 00:28:00.470 is telling us is the following. 00:28:00.470 --> 00:28:04.730 You take a set of random points, and you project them 00:28:04.730 --> 00:28:06.070 into higher dimensions. 00:28:06.070 --> 00:28:08.700 You start with 100 dimensions, and you project them 00:28:08.700 --> 00:28:10.820 in 1,000 dimensions. 00:28:10.820 --> 00:28:12.870 On average, they are random. 00:28:12.870 --> 00:28:16.395 But actually-- so you would imagine 00:28:16.395 --> 00:28:17.990 that it's a perfect thing. 00:28:17.990 --> 00:28:20.870 You project them with random weights. 00:28:20.870 --> 00:28:22.490 Then you would imagine that you just 00:28:22.490 --> 00:28:28.240 created a set of random points in the expanded dimension 00:28:28.240 --> 00:28:30.030 representation. 00:28:30.030 --> 00:28:34.160 If this was so, then if you do SVM or PCA 00:28:34.160 --> 00:28:36.830 on this representation, you will find 00:28:36.830 --> 00:28:40.190 what you expect from a PCA of a set of random points. 00:28:40.190 --> 00:28:42.050 And this is this one. 00:28:42.050 --> 00:28:50.250 In fact, there is a trace of low dimensionality in the data. 00:28:50.250 --> 00:28:52.620 So I think that's an important point, which 00:28:52.620 --> 00:28:53.930 I would like to explain. 00:28:53.930 --> 00:28:56.450 You start from a set of points. 00:28:56.450 --> 00:29:00.410 If you don't threshold them and you just map them 00:29:00.410 --> 00:29:06.710 into 1,000-dimensional space, those 100-dimensional input 00:29:06.710 --> 00:29:08.600 will remain 100-dimensional. 00:29:08.600 --> 00:29:10.820 Just be rotated, and so on, but everything 00:29:10.820 --> 00:29:14.970 will live in 100-dimensional space. 00:29:14.970 --> 00:29:20.300 Now you add thresholding, high thresholding by sparsity. 00:29:20.300 --> 00:29:24.770 So those 100-dimensional subspace becomes now 00:29:24.770 --> 00:29:28.900 1,000-dimensional sparsity because of the nonlinearity. 00:29:28.900 --> 00:29:33.740 But this nonlinearity, although it takes 100-dimensional input 00:29:33.740 --> 00:29:36.560 and makes them 1,000-dimensional, 00:29:36.560 --> 00:29:40.040 it's still not like random. 00:29:40.040 --> 00:29:45.230 This 1,000-dimensional cloud is still elongated. 00:29:45.230 --> 00:29:47.630 It's not simply uniformly distributed. 00:29:47.630 --> 00:29:52.400 And this is the signature that you see here. 00:29:52.400 --> 00:29:55.130 In the first largest 100 eigenvalues, 00:29:55.130 --> 00:30:02.870 there is extra power relative to the random. 00:30:02.870 --> 00:30:04.040 The rest is not 0. 00:30:04.040 --> 00:30:07.220 So if you look here, this goes up to 1,000. 00:30:07.220 --> 00:30:09.860 The rest is not 0. 00:30:09.860 --> 00:30:13.580 So the system is, strictly speaking, 00:30:13.580 --> 00:30:17.370 1,000-dimensional space, but it's not random. 00:30:17.370 --> 00:30:22.370 It has increased power in 100 channels. 00:30:22.370 --> 00:30:26.460 If you do a readout, a linear classifier 00:30:26.460 --> 00:30:29.880 readout, what you find in this-- 00:30:29.880 --> 00:30:39.860 again, when you expand with random weights, 00:30:39.860 --> 00:30:45.120 you find that there is an optimal sparsity. 00:30:45.120 --> 00:30:46.920 So this is the readout error. 00:30:46.920 --> 00:30:51.326 For a classifier, the output is a function 00:30:51.326 --> 00:30:54.920 of the sparsity for different levels of noise. 00:30:54.920 --> 00:30:59.960 And you see that, in the case of random weights, 00:30:59.960 --> 00:31:05.470 there is a very high sparsity, is bad. 00:31:05.470 --> 00:31:08.270 There Is an optimal sparsity or sparseness, 00:31:08.270 --> 00:31:10.820 and then there is a shallow increase 00:31:10.820 --> 00:31:14.900 in the error when you go to a denser representation. 00:31:14.900 --> 00:31:18.530 One important point which I want to emphasize 00:31:18.530 --> 00:31:21.000 coming from the analysis-- let me skip equations-- 00:31:21.000 --> 00:31:22.580 and this is what you see here. 00:31:22.580 --> 00:31:28.070 The question is, can I do better by further increase the layer? 00:31:28.070 --> 00:31:32.090 So here I plot the readout error as a function 00:31:32.090 --> 00:31:34.400 of the size of cortical layer. 00:31:34.400 --> 00:31:36.410 Can I do better? 00:31:36.410 --> 00:31:39.830 If I make the kernel dimensionality infinite, 00:31:39.830 --> 00:31:41.690 can I do better? 00:31:41.690 --> 00:31:44.760 Well, it can do better if you start with 0 noise. 00:31:44.760 --> 00:31:52.265 But if you have noisy inputs, then basically, 00:31:52.265 --> 00:31:54.140 the performance saturates. 00:31:54.140 --> 00:31:55.490 And that's kind of surprising. 00:31:58.670 --> 00:32:00.890 We were expecting that, if you go 00:32:00.890 --> 00:32:04.382 to a larger and larger representation, eventually 00:32:04.382 --> 00:32:05.780 the error will go to 0. 00:32:05.780 --> 00:32:06.920 But it doesn't go to 0. 00:32:06.920 --> 00:32:09.980 And that actually happens even for what we 00:32:09.980 --> 00:32:12.590 call structured representation. 00:32:12.590 --> 00:32:15.560 And that's the same for different types of readout-- 00:32:15.560 --> 00:32:18.830 perceptual, and pseudo-inverse, SVM. 00:32:18.830 --> 00:32:22.040 All of them show this saturation as you increase 00:32:22.040 --> 00:32:24.120 the size of cortical layer. 00:32:24.120 --> 00:32:29.110 And that's one of the very important outcome of our study. 00:32:29.110 --> 00:32:31.740 That when you talk about noisy inputs, 00:32:31.740 --> 00:32:35.590 you can think about it as kind of more generalization task. 00:32:35.590 --> 00:32:40.580 Then there is a limit about what you gain 00:32:40.580 --> 00:32:43.105 by expanding representation. 00:32:43.105 --> 00:32:46.120 Even if you expand in a nonlinear fashion and you 00:32:46.120 --> 00:32:50.440 increase the dimensionality, you cannot combat the noise, 00:32:50.440 --> 00:32:52.040 at least up to some level. 00:32:52.040 --> 00:32:55.340 Beyond some level, there is no point of further expansion, 00:32:55.340 --> 00:33:02.190 because basically the error saturates 00:33:02.190 --> 00:33:05.220 Let me, since time goes fast, let 00:33:05.220 --> 00:33:06.540 me talk about the alternatives. 00:33:06.540 --> 00:33:09.167 So if random weights are not doing so well, 00:33:09.167 --> 00:33:10.250 what are the alternatives? 00:33:10.250 --> 00:33:14.760 The alternative is to do some kind of unsupervised learning. 00:33:14.760 --> 00:33:18.300 Here we are doing it in a kind of a shortcut 00:33:18.300 --> 00:33:20.230 of unsupervised learning. 00:33:20.230 --> 00:33:21.120 What is the shortcut? 00:33:21.120 --> 00:33:22.810 We say the following. 00:33:22.810 --> 00:33:26.880 Imagine that these layers, the learner 00:33:26.880 --> 00:33:31.130 knows about the representation of the clusters. 00:33:31.130 --> 00:33:33.350 It doesn't know the labels. 00:33:33.350 --> 00:33:35.475 In other words, whether those are pluses and those 00:33:35.475 --> 00:33:38.130 are minuses, which one are pluses and minuses. 00:33:38.130 --> 00:33:40.860 But he does know about the statistical structure 00:33:40.860 --> 00:33:43.740 of the input, and this is this S bar. 00:33:43.740 --> 00:33:45.900 These are the centers. 00:33:45.900 --> 00:33:49.860 So we want to encode the statistical structure 00:33:49.860 --> 00:33:54.660 of these input in this expansion of the weights. 00:33:54.660 --> 00:33:57.840 And the way we do the simplest way is the kind of Hebb rule. 00:33:57.840 --> 00:33:58.870 We do the following. 00:33:58.870 --> 00:34:05.150 We say let's first choose, or recruit, or allocate a state, 00:34:05.150 --> 00:34:10.530 a sparse state here, randomly chosen, to associate, 00:34:10.530 --> 00:34:13.120 to represent each one of the clusters. 00:34:13.120 --> 00:34:18.150 So these are the R. R are the randomly chosen patterns here. 00:34:18.150 --> 00:34:22.239 And then we associate between those randomly chosen 00:34:22.239 --> 00:34:27.090 representations and the actual centers 00:34:27.090 --> 00:34:28.560 of the clusters of the inputs. 00:34:28.560 --> 00:34:31.880 So this is S bar and R. And then we do the association 00:34:31.880 --> 00:34:34.440 by the simple, what's called Hebb rule. 00:34:34.440 --> 00:34:37.380 So this Hebbian rule associates cluster 00:34:37.380 --> 00:34:43.920 center with a randomly assigned state 00:34:43.920 --> 00:34:49.860 in the cortical layer in a kind of simple summation 00:34:49.860 --> 00:34:51.630 or outer product for the Hebb rule. 00:34:51.630 --> 00:34:53.580 There are more sophisticated ways to do it, 00:34:53.580 --> 00:34:56.280 but that's the simplest one of doing it. 00:34:56.280 --> 00:35:03.090 So it turns out that this simple rule has enormous potential 00:35:03.090 --> 00:35:04.820 for suppressing noise. 00:35:04.820 --> 00:35:07.830 So, again, this is the input noise and the output noise. 00:35:07.830 --> 00:35:10.830 The Hamming distance of the input and the output 00:35:10.830 --> 00:35:12.240 properly normalized. 00:35:12.240 --> 00:35:16.740 And you see that, as you go to higher and higher sparseness, 00:35:16.740 --> 00:35:20.555 to lower and lower f, this is basically 00:35:20.555 --> 00:35:24.930 the input noise is completely quenched when f is large. 00:35:24.930 --> 00:35:28.700 When f is 0.01, for instance, this is this already. 00:35:28.700 --> 00:35:33.120 Sub-linear when f 0.05 is here, and so and so forth. 00:35:33.120 --> 00:35:36.195 So sparse representation, in particular, 00:35:36.195 --> 00:35:39.580 are very effective in suppressing noise, 00:35:39.580 --> 00:35:45.210 but provided the inputs have kind of unsupervised learning 00:35:45.210 --> 00:35:48.750 encoded into them which embed into them 00:35:48.750 --> 00:35:51.361 the cluster structure of the inputs. 00:35:55.740 --> 00:35:58.170 The same or similar thing is true for Q 00:35:58.170 --> 00:35:59.910 for these correlations. 00:35:59.910 --> 00:36:03.630 If you look at the-- this was a random correlation. 00:36:03.630 --> 00:36:08.520 This is a function of f, and this is Q, the correlation. 00:36:08.520 --> 00:36:12.840 It's extremely suppressed for sparse representation. 00:36:12.840 --> 00:36:15.540 Basically, it's exponentially small with 1/f, 00:36:15.540 --> 00:36:18.660 so it's basically 0 for sparse representation. 00:36:18.660 --> 00:36:21.780 Which means that those centers look like randomly 00:36:21.780 --> 00:36:25.200 distributed, essentially, and with very small noise. 00:36:25.200 --> 00:36:28.700 So you took these spheres and you basically 00:36:28.700 --> 00:36:33.570 map them into random points with a very small radius. 00:36:33.570 --> 00:36:36.610 So it's not surprising that, in this case, 00:36:36.610 --> 00:36:42.670 the error for small f-- the error, even for large noise 00:36:42.670 --> 00:36:46.700 values, the error is basically small, 0. 00:36:46.700 --> 00:36:48.750 Nevertheless, it is still saturating 00:36:48.750 --> 00:36:52.500 as a function of the network size, of the cortical size. 00:36:52.500 --> 00:36:57.520 So the saturation of performance as a function of cortical size 00:36:57.520 --> 00:37:00.910 is a general property of such systems. 00:37:00.910 --> 00:37:05.400 Nevertheless, the performance itself for any given size 00:37:05.400 --> 00:37:13.830 is extremely impressive, I would say, when the system is sparse 00:37:13.830 --> 00:37:16.852 and the noise level is kind of moderate. 00:37:20.370 --> 00:37:22.530 OK, let me skip this because I don't have time. 00:37:22.530 --> 00:37:27.570 Let me briefly talk about extension of this story 00:37:27.570 --> 00:37:28.650 to multi-layer. 00:37:28.650 --> 00:37:31.710 So we are now briefly discussing what 00:37:31.710 --> 00:37:34.800 happens if you take this story and you just propagate it 00:37:34.800 --> 00:37:38.662 as you go along the architecture. 00:37:38.662 --> 00:37:40.120 So let's start with random weights. 00:37:40.120 --> 00:37:43.140 So the idea is maybe something is good happening. 00:37:43.140 --> 00:37:45.610 Although initially performance was poor, 00:37:45.610 --> 00:37:47.250 maybe we can improve the performance 00:37:47.250 --> 00:37:50.160 by cascading such layers. 00:37:50.160 --> 00:37:53.010 And the answer is no, particularly the noise level. 00:37:53.010 --> 00:37:55.950 This is now the number of layers. 00:37:55.950 --> 00:37:57.840 What we discussed before is here. 00:37:57.840 --> 00:38:00.120 And you see the problem becomes worse and worse. 00:38:00.120 --> 00:38:04.350 As you continue to propagate those signals, 00:38:04.350 --> 00:38:08.050 the noise is amplified and essentially goes to 1. 00:38:08.050 --> 00:38:12.000 So basically you will get just random performance 00:38:12.000 --> 00:38:16.420 if you keep doing it with random weights. 00:38:16.420 --> 00:38:20.520 The reason-- where is it? 00:38:20.520 --> 00:38:21.600 I missed a slide. 00:38:21.600 --> 00:38:23.520 The reason is, basically, that if you 00:38:23.520 --> 00:38:28.380 think about the mapping from one layer of noise 00:38:28.380 --> 00:38:31.620 to another layer of noise, there are two fixed points, 0 and 1. 00:38:31.620 --> 00:38:33.570 The 0 fixed point is unstable. 00:38:33.570 --> 00:38:35.490 Everything goes eventually to 1. 00:38:35.490 --> 00:38:39.090 So it is a nice-- 00:38:39.090 --> 00:38:41.340 this system gives you a nice perspective 00:38:41.340 --> 00:38:48.390 about this deep network by thinking about it 00:38:48.390 --> 00:38:49.780 as a kind of dynamical system. 00:38:49.780 --> 00:38:51.680 For instance, what is the level of noise 00:38:51.680 --> 00:38:56.520 at one layer, how it's related to the level of noise 00:38:56.520 --> 00:38:57.480 at previous layer. 00:38:57.480 --> 00:38:59.610 So it's kind of iterative map. 00:38:59.610 --> 00:39:01.950 Delta n versus delta n minus 1. 00:39:01.950 --> 00:39:05.040 And what's good about it is, once you kind of draw 00:39:05.040 --> 00:39:08.100 this curve, one layer is mapped to another layer, 00:39:08.100 --> 00:39:10.200 you can know what happens to a deep network. 00:39:10.200 --> 00:39:11.769 We could just iterate this. 00:39:11.769 --> 00:39:13.560 You have to find what are the fixed points, 00:39:13.560 --> 00:39:15.434 and which one is stable and which one is not. 00:39:15.434 --> 00:39:18.970 In this case, the 1 is stable, the 0 is unstable. 00:39:18.970 --> 00:39:21.120 So, unfortunately, from any level of noise 00:39:21.120 --> 00:39:24.390 that you will start, you eventually go to 1. 00:39:27.582 --> 00:39:29.750 Correlations, is a similar story, but-- 00:39:29.750 --> 00:39:33.370 and the error will go to 0.5. 00:39:33.370 --> 00:39:36.610 So that's very well. 00:39:36.610 --> 00:39:40.270 There are cases, by the way, that you can find parameters 00:39:40.270 --> 00:39:43.000 where initially you improve, like here. 00:39:43.000 --> 00:39:47.290 But then eventually it will go to 0.5. 00:39:47.290 --> 00:39:49.720 Now, if we do similar-- 00:39:49.720 --> 00:39:51.790 if we compare this to what happened 00:39:51.790 --> 00:39:53.800 to the structured weights if you keep 00:39:53.800 --> 00:39:57.490 doing the same kind of unsupervised Hebbian 00:39:57.490 --> 00:40:00.700 learning from one layer to another-- 00:40:00.700 --> 00:40:03.970 and I'll skip the details-- you see the opposite. 00:40:03.970 --> 00:40:08.140 So here are parameter value in which 00:40:08.140 --> 00:40:14.470 one stage of the expansion stage is actually 00:40:14.470 --> 00:40:15.550 increasing the noise. 00:40:15.550 --> 00:40:21.580 And this is because f is not too small, and the load is large, 00:40:21.580 --> 00:40:23.540 and the noise is starting. 00:40:23.540 --> 00:40:25.360 So you can have such situation. 00:40:25.360 --> 00:40:28.390 But even in such situation, eventually the system 00:40:28.390 --> 00:40:34.855 goes into stages where the noise basically goes to 0. 00:40:34.855 --> 00:40:38.380 And if you compare the story why it 00:40:38.380 --> 00:40:42.230 is so to kind of iterative map picture, 00:40:42.230 --> 00:40:44.210 you see that the picture is very different. 00:40:44.210 --> 00:40:45.980 You have one fixed point at 0. 00:40:45.980 --> 00:40:47.740 You have one fixed point at 1. 00:40:47.740 --> 00:40:51.010 You have intermediate fixed point at high value. 00:40:51.010 --> 00:40:53.550 But this is an unstable fixed point, and both of them 00:40:53.550 --> 00:40:54.550 are stable fixed points. 00:40:54.550 --> 00:40:58.540 So if you start from even from large values of noise, 00:40:58.540 --> 00:41:00.700 eventually you will iterate to 0. 00:41:00.700 --> 00:41:05.290 So it does buy you to actually go 00:41:05.290 --> 00:41:10.120 into several stages of this deep network to make sure 00:41:10.120 --> 00:41:12.730 that the noise is suppressed to 0. 00:41:12.730 --> 00:41:14.380 Similarly for the correlations. 00:41:14.380 --> 00:41:17.440 Even if the parameters are such that initially correlations 00:41:17.440 --> 00:41:20.350 are increased, and you can find parameters like that, 00:41:20.350 --> 00:41:22.720 eventually correlations will go to almost 0. 00:41:25.360 --> 00:41:28.840 And this is comparison of the readout error 00:41:28.840 --> 00:41:32.370 as a function of the layers with structured weights, 00:41:32.370 --> 00:41:34.840 and I compare it with the readout error 00:41:34.840 --> 00:41:38.740 of infinitely wide layer, kind of a kernel 00:41:38.740 --> 00:41:41.710 with infinitely wide kernel. 00:41:41.710 --> 00:41:44.830 And you can see that, at least for-- 00:41:44.830 --> 00:41:49.240 here I compare the same type of unsupervised learning but two 00:41:49.240 --> 00:41:50.290 different architectures. 00:41:50.290 --> 00:41:52.270 One is deep network architecture, 00:41:52.270 --> 00:41:57.180 and the another one is shallow architecture, infinitely wide. 00:41:57.180 --> 00:42:01.720 I'm not claiming that we can show that there is no kernel 00:42:01.720 --> 00:42:03.940 or shallow architecture which will do better, 00:42:03.940 --> 00:42:08.650 but I'm saying if we compare the same learning rule 00:42:08.650 --> 00:42:11.050 but with the two different architectures, 00:42:11.050 --> 00:42:14.320 you'll find that you do gain by going 00:42:14.320 --> 00:42:17.980 into multiple stages of nonlinearity 00:42:17.980 --> 00:42:20.590 than by using an infinitely wide layer. 00:42:24.210 --> 00:42:24.980 I'll skip this. 00:42:24.980 --> 00:42:31.070 I want to go briefly to two more issues. 00:42:31.070 --> 00:42:34.460 One issue is the recurrent networks. 00:42:34.460 --> 00:42:36.200 Why recurrent networks? 00:42:36.200 --> 00:42:39.830 The primary reason is that, in each one 00:42:39.830 --> 00:42:43.860 of those stages that I refer to, if you look at the biology, 00:42:43.860 --> 00:42:45.590 on most of them-- 00:42:45.590 --> 00:42:48.795 not all of them but most of them, and definitely 00:42:48.795 --> 00:42:50.150 in neocortex-- 00:42:50.150 --> 00:42:53.810 you find massive recurrent or lateral interactions 00:42:53.810 --> 00:42:56.400 between each one of the layers. 00:42:56.400 --> 00:43:01.700 So, again, we would like to ask, what 00:43:01.700 --> 00:43:06.560 is the computational advantage of having this recurrent layer. 00:43:06.560 --> 00:43:11.090 Now, in our case, we had an extra motivation, and this is-- 00:43:11.090 --> 00:43:16.140 remember that I started in saying that, in some cases, 00:43:16.140 --> 00:43:19.550 there is experimental evidence that the initial projection is 00:43:19.550 --> 00:43:21.710 random. 00:43:21.710 --> 00:43:24.170 So that we ask ourselves, what happens if we do this. 00:43:24.170 --> 00:43:26.466 If we start from random projection, 00:43:26.466 --> 00:43:31.160 feedforward projection, and then add recurrent connections. 00:43:31.160 --> 00:43:34.560 Think about it as from the olfactory bulb, for instance, 00:43:34.560 --> 00:43:38.470 to piriform cortex, perhaps random feedforward projections. 00:43:38.470 --> 00:43:42.235 But then the association, recurrent connections 00:43:42.235 --> 00:43:45.200 in piriform cortex are structured. 00:43:45.200 --> 00:43:46.560 How do we do that? 00:43:46.560 --> 00:43:51.020 We start, we imagine starting from random projection, 00:43:51.020 --> 00:43:54.290 generating initial representation 00:43:54.290 --> 00:43:57.110 by the random projection, and then stabilizing 00:43:57.110 --> 00:43:58.910 those representation into attractors 00:43:58.910 --> 00:44:00.590 by the recurrent connections. 00:44:00.590 --> 00:44:04.780 And that actually works pretty well. 00:44:04.780 --> 00:44:06.680 It's not the optimal architecture, 00:44:06.680 --> 00:44:07.700 but it's pretty well. 00:44:07.700 --> 00:44:10.850 For instance, noise, which is initially 00:44:10.850 --> 00:44:13.010 increased by the random projections, 00:44:13.010 --> 00:44:16.510 were quenched by convergence to attractors. 00:44:16.510 --> 00:44:18.900 And, similarly, Q will not go to 0, 00:44:18.900 --> 00:44:20.810 but will not continue growing, but will 00:44:20.810 --> 00:44:23.150 go to an intermediate layer. 00:44:23.150 --> 00:44:25.760 And the error is pretty well. 00:44:25.760 --> 00:44:30.260 So if you look at in this case, the error really 00:44:30.260 --> 00:44:32.730 goes down to very low values. 00:44:32.730 --> 00:44:34.190 But now it's not layers. 00:44:34.190 --> 00:44:36.440 Now it is the number of iterations 00:44:36.440 --> 00:44:38.150 of the recurrent connections. 00:44:38.150 --> 00:44:44.000 So you start from just input layer, or random projection, 00:44:44.000 --> 00:44:47.450 and then you iterate the dynamics and it goes to 0. 00:44:47.450 --> 00:44:48.420 So it's not the layers. 00:44:48.420 --> 00:44:55.160 It's just the dynamics of the convergence to attractor. 00:44:55.160 --> 00:44:56.570 My final point. 00:44:56.570 --> 00:44:58.600 I have 3 or 4 minutes? 00:44:58.600 --> 00:44:59.300 OK. 00:44:59.300 --> 00:45:04.670 My final point before wrapping up is the question of top-down. 00:45:04.670 --> 00:45:09.980 So recurrent, we briefly talked about it. 00:45:09.980 --> 00:45:13.790 But incorporating contextual knowledge is a major question. 00:45:13.790 --> 00:45:17.900 How can you improve on deep networks 00:45:17.900 --> 00:45:22.820 by incorporating, not simply the feedforward sensory input, 00:45:22.820 --> 00:45:28.400 but other sources of knowledge about this particular stimulus? 00:45:28.400 --> 00:45:33.080 And it's important that we are not talking about knowledge 00:45:33.080 --> 00:45:35.690 about the statistics of the input which 00:45:35.690 --> 00:45:37.730 can be incorporated into the learning 00:45:37.730 --> 00:45:39.200 of the feedforward one. 00:45:39.200 --> 00:45:42.450 But we're talking about inputs which are, 00:45:42.450 --> 00:45:47.330 or knowledge, which we have now on the network which already 00:45:47.330 --> 00:45:49.430 has learned whatever it has learned. 00:45:49.430 --> 00:45:52.610 So we have a mature network, whatever the architecture is. 00:45:52.610 --> 00:45:53.840 We have a sensory input. 00:45:53.840 --> 00:45:55.400 It goes feedforward. 00:45:55.400 --> 00:45:58.610 And now we have additional information, about 00:45:58.610 --> 00:46:00.170 context for instance, that we want 00:46:00.170 --> 00:46:03.140 to incorporate with the sensory input 00:46:03.140 --> 00:46:05.345 to improve the performance. 00:46:05.345 --> 00:46:09.260 So how do we do that? 00:46:09.260 --> 00:46:13.460 It turns out to be non-trivial computational problem. 00:46:13.460 --> 00:46:18.980 It is very straightforward to do it in Bayesian framework, 00:46:18.980 --> 00:46:21.550 where you simply update the prior 00:46:21.550 --> 00:46:29.070 of what the sensory input is by this contextual information. 00:46:29.070 --> 00:46:32.210 But if you want to implement it in the network, 00:46:32.210 --> 00:46:36.910 you find that it's not easy to find 00:46:36.910 --> 00:46:38.960 the appropriate architecture. 00:46:38.960 --> 00:46:43.530 So I'll just briefly talk about how we do it. 00:46:43.530 --> 00:46:46.670 So imagine you have, again, these sensory inputs, 00:46:46.670 --> 00:46:50.460 but now there is some context, different contexts. 00:46:50.460 --> 00:46:53.840 And imagine you have an information 00:46:53.840 --> 00:47:00.710 that the input is coming from that particular part of state 00:47:00.710 --> 00:47:02.040 space. 00:47:02.040 --> 00:47:05.030 So basically the question is how to amplify selectively 00:47:05.030 --> 00:47:08.630 a specific set of states in a distributed representation. 00:47:08.630 --> 00:47:12.790 So usually when we talk about attention, or gating, 00:47:12.790 --> 00:47:15.020 or questions like that, we think about, OK, we 00:47:15.020 --> 00:47:16.640 have these neurons. 00:47:16.640 --> 00:47:20.650 We suppress those, or maybe amplify other ones. 00:47:20.650 --> 00:47:24.310 Or we have a set of axons, or pathways. 00:47:24.310 --> 00:47:26.890 We suppress those, and amplify those. 00:47:26.890 --> 00:47:29.410 But what about a representation which is more 00:47:29.410 --> 00:47:33.040 distributed where you have to really suppress states 00:47:33.040 --> 00:47:36.730 rather than neural populations. 00:47:36.730 --> 00:47:42.850 So I just won't go-- again, it's a complicated architecture. 00:47:42.850 --> 00:47:48.010 But, basically, we're using some sort of a mixed representation, 00:47:48.010 --> 00:47:52.090 where we take the sensory input and the category 00:47:52.090 --> 00:47:55.510 or contextual input, mix the nonlinearity, 00:47:55.510 --> 00:47:58.614 use them to clean it, and propagate this. 00:47:58.614 --> 00:48:00.280 So it's a more complicated architecture, 00:48:00.280 --> 00:48:01.960 but it works beautifully. 00:48:01.960 --> 00:48:04.150 Let me show you here an example, and you'll have 00:48:04.150 --> 00:48:05.770 a flavor of what we are doing. 00:48:05.770 --> 00:48:13.420 So now the input, we have those 900 spheres or templates, 00:48:13.420 --> 00:48:20.110 but they are organized into 30 categories, 00:48:20.110 --> 00:48:23.440 and 30 tokens per category. 00:48:23.440 --> 00:48:27.400 Now, the tokens, which are the actual sensory inputs, 00:48:27.400 --> 00:48:30.880 are represented by, let's say, 200 neurons. 00:48:30.880 --> 00:48:32.695 And you have a small number of neurons 00:48:32.695 --> 00:48:34.230 representing a category. 00:48:34.230 --> 00:48:35.434 Maybe 20 is enough. 00:48:35.434 --> 00:48:36.850 So that's important, and you don't 00:48:36.850 --> 00:48:39.320 have to really expand dramatically 00:48:39.320 --> 00:48:42.430 the representation. 00:48:42.430 --> 00:48:45.140 So this is the input. 00:48:45.140 --> 00:48:48.340 And now we have very noisy inputs. 00:48:48.340 --> 00:48:51.440 If you look at the readout, this is layers here, 00:48:51.440 --> 00:48:52.600 and there is readout error. 00:48:52.600 --> 00:48:57.140 If you do it on the input layer, or any subsequent layer 00:48:57.140 --> 00:49:00.290 here, but without top-down information. 00:49:00.290 --> 00:49:03.610 With structured interactions and all that I told you, 00:49:03.610 --> 00:49:07.420 this is such a noisy input where the performance is basically 00:49:07.420 --> 00:49:08.800 0.5. 00:49:08.800 --> 00:49:13.150 There is nothing that you can do without top-down information 00:49:13.150 --> 00:49:14.540 in this network. 00:49:14.540 --> 00:49:17.600 You can ask what will be the performance. 00:49:17.600 --> 00:49:21.910 If you have an ideal observer that looks at the noisy input 00:49:21.910 --> 00:49:27.310 and makes maximum likelihood categorization. 00:49:27.310 --> 00:49:28.920 Well, then it will do much better. 00:49:28.920 --> 00:49:31.390 Also not 0, so this is at this level. 00:49:33.970 --> 00:49:38.650 This higher error is in virtue of the fact 00:49:38.650 --> 00:49:42.190 that this network is still not doing 00:49:42.190 --> 00:49:46.990 what an optimal maximum likelihood observer will do. 00:49:46.990 --> 00:49:48.320 So this is the network. 00:49:48.320 --> 00:49:53.110 This is a maximum likelihood readout, both of them 00:49:53.110 --> 00:49:56.320 without extra top-down information. 00:49:56.320 --> 00:49:59.950 And in the network that I kind of hinted about, 00:49:59.950 --> 00:50:04.570 if you add this top-down information by generating 00:50:04.570 --> 00:50:08.740 mixed representation, you get a performance which is really 00:50:08.740 --> 00:50:11.210 dramatically improved. 00:50:11.210 --> 00:50:16.690 And as you keep doing it one layer from another, 00:50:16.690 --> 00:50:21.310 you really get a very nice performance. 00:50:21.310 --> 00:50:24.010 So let me just summarize. 00:50:28.490 --> 00:50:31.735 There is one more before summarizing. 00:50:31.735 --> 00:50:33.130 Yeah, OK. 00:50:33.130 --> 00:50:33.820 Before that. 00:50:33.820 --> 00:50:34.320 OK. 00:50:34.320 --> 00:50:40.930 So two points to bear in mind. 00:50:40.930 --> 00:50:44.790 One of them is that what I discussed to you today 00:50:44.790 --> 00:50:51.460 relies on assuming either random, comparing random 00:50:51.460 --> 00:50:56.410 projection, to unsupervised learning of a very simple type, 00:50:56.410 --> 00:50:59.010 of a kind of Hebbian type. 00:50:59.010 --> 00:51:07.480 The output can be Hebbian, or perceptron, or SVM, and so on. 00:51:07.480 --> 00:51:09.040 You could ask, what happens if you 00:51:09.040 --> 00:51:12.850 use learning rules, more sophisticated learning rules 00:51:12.850 --> 00:51:14.140 for the unsupervised weights? 00:51:14.140 --> 00:51:15.280 Some of them we've studied. 00:51:15.280 --> 00:51:20.470 But, anyway, that's something which is important to explore. 00:51:20.470 --> 00:51:24.640 And another very important issue for thinking 00:51:24.640 --> 00:51:28.110 about object recognition in vision 00:51:28.110 --> 00:51:33.340 and in other real-life problem is input statistics. 00:51:33.340 --> 00:51:36.070 Because what we assumed is a very simple mixture 00:51:36.070 --> 00:51:37.870 of Gaussian model. 00:51:37.870 --> 00:51:40.930 So you can think about the task of the network 00:51:40.930 --> 00:51:46.270 is to take the invariance, which is the variation away 00:51:46.270 --> 00:51:49.080 from the center of the spherical variation, 00:51:49.080 --> 00:51:52.870 and to generate representation which is invariant to that. 00:51:52.870 --> 00:51:55.510 But this is a very simple invariance problem, 00:51:55.510 --> 00:51:58.450 because the invariance was simply 00:51:58.450 --> 00:52:03.790 restricted to these simple geometric structures. 00:52:03.790 --> 00:52:12.070 More problems which are closer to what real-life problems are 00:52:12.070 --> 00:52:17.200 will have inputs which are, essentially, have 00:52:17.200 --> 00:52:19.240 some structure, but the structure 00:52:19.240 --> 00:52:22.860 can be of a variety of shapes. 00:52:22.860 --> 00:52:27.090 Each one of them correspond to an object, or a cluster, 00:52:27.090 --> 00:52:31.850 or a manifold representing an entity, a perceptual entity. 00:52:31.850 --> 00:52:37.220 But how you go from this nice, simple problem 00:52:37.220 --> 00:52:43.100 of this spherical invariance problem to those problems, 00:52:43.100 --> 00:52:45.780 it's of course a challenging problem. 00:52:45.780 --> 00:52:50.300 And that's the work which we are now, ongoing work, also 00:52:50.300 --> 00:52:53.990 with SueYeon Chung and Dan Lee. 00:52:53.990 --> 00:53:00.130 But it's a story which is still at the stage of unfolding.