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E ﬃ cient Coding: From Retina Ganglion Cells To V2 Cells Honghao Shan Garrison W. Cottrell The Temporal Dynamics of Learning Center Gary's Unbelievable Research Unit (GURU) Computer Science and Engineering Department Institute for Neural Computation UCSD

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Introduction and Motivation We have 10 11 − 10 12 neurons with ~10 15 connections between them - it seems highly unlikely that the features they respond to are learned by any supervised mechanism! Hence unsupervised learning seems much more likely. What is the correct learning rule? Here we focus on the visual system.

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Introduction and Motivation

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The retina has about 100 million photoreceptors These are processed into about 1 million ganglion cell outputs These cells have a receptive field, which is an overloaded term to mean both 1. the region of space that they are sensitive to, and 2. the pattern that region must contain to fire them.

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Introduction and Motivation Ganglion cell receptive fields are ON-center, OFF- surround, and vice-versa: ON-CENTEROFF-CENTER Stimulus Response When stimulus is on

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Introduction and Motivation One of the consequences of these cells:

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Introduction and Motivation In V1, simple cells respond to oriented visual edges In V1, complex cells respond to visual edges at nearby locations - they appear to pool the responses of simple cells In V2, cell responses are already hard to characterize. Eventually, there are cells that respond to faces, and even further in, respond to identity (faces and names).

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Introduction and Motivation E.g., the “Halle Berry” neuron…

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Introduction and Motivation If these are learned by unsupervised learning, then what is the correct learning rule? What is the goal of the learning rule? Hypothesis: visual perception serves to capture statistical structure of the visual inputs Attneave (1954): the statistical structure can be measured by the redundancy of the inputs: I(x) = i H(x i ) − H(x) (Minimized to zero when x i are independent). the efficient coding theory Barlow (1961) suggested what has come to be called the efficient coding theory: the goal of early vision is to remove redundancy from the visual inputs. The coding (outputs) should be as independent as possible

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Introduction and Motivation There have been a variety of implementations of the efficient coding theory: Principal Components Analysis (PCA): provably optimal (in a least squares sense) linear dimensionality technique

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Introduction and Motivation Principal Components Analysis (PCA) - but this only leads to uncorrelated outputs, and global receptive fields that look nothing like V1 receptive fields.

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Introduction and Motivation There have been a variety of implementations of the efficient coding theory: Principal Components Analysis (PCA) - but this only leads to uncorrelated outputs, and global receptive fields Independent Components Analysis (ICA) (Bell & Sejnowski) Sparse Coding (Olshausen & Field) These last two: Lead to Gabor like receptive fields (as we see in V1) Turn out to be equivalent under certain assumptions.

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Introduction and Motivation When applied to natural images, ICA and sparse coding give Gabor-like receptive fields, similar to those seen in simple cells in V1

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Introduction and Motivation There have been many attempts to go beyond a single layer (Karklin, Y., & Lewicki, M. S. (2006), Schwartz, O., & Simoncelli, E. P. (2001) Hoyer & Hyvarinen (2002) but: They usually require a different learning rule And do not lead to a way to do the next layer. Or, like deep belief networks, do not have plausible receptive fields (Hinton, 2006)

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Our contribution We have previously developed a method for applying ICA over and over, in order to get higher layer representations (Shan, Zhang, & Cottrell, NIPS, 2006/2007), called RICA Recursive Independent Components Analysis: ICA->add nonlinearity->ICA->add nonlinearity… In our paper, we showed that the second layer of ICA had interesting neural properties

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Roadmap We describe ICA and our version of a hierarchical ICA, Recursive ICA (RICA) We illustrate PCA and describe sparse PCA (SPCA) and the initial results We investigate the receptive fields of the higher layers of RICA.

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Roadmap We describe ICA and our version of a hierarchical ICA, Recursive ICA (RICA) We illustrate PCA and describe sparse PCA (SPCA) and the initial results We investigate the receptive fields of the higher layers of RICA.

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Independent Components Analysis Sparse coding and ICA can be summarized by a generative model, in which the observed data vector x ∈ R L is assumed to be a linear mixture of some (unknown) underlying signal sources s ∈ R M : the pixels ->x = As + ε <- s is the neural firings where A ∈ R LXM denotes the (unknown) mixing matrix, ε ∈ R L denotes additive Gaussian noise I.e., the pixels are a linear combination of neural firings (underlying signal sources - what we believe is in the input).

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Independent Components Analysis x = As + ε Two assumptions are imposed on the signal sources: 1. They are assumed to be independent 2. Each dimension of s is assumed to follow a sparse distribution, usually with a peak at zero and two heavy tails, unlike PCA, where the projections tend to follow a Gaussian. Model parameters are adapted to make the generation of the observed x’s likely, and to encourage the sparse prior:

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Recursive Independent Components Analysis (RICA 1.0) We assume that higher layers of cortex follow a similar learning rule as earlier layers - so we would like to apply essentially the same learning rule to subsequent layers. BUT: More linear processing will not lead to more interesting structure… So we need some nonlinearity applied to the output of the first layer of ICA…

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Recursive Independent Components Analysis (RICA 1.0) Notice that the generative model: x = As + ε means that: x i = j A ij *s j + ε i Thus, each input variable (pixel) is assumed to be the sum of many independent random variables…

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Recursive Independent Components Analysis (RICA 1.0) Thus, each input variable (pixel) is assumed to be the sum of many independent random variables… i.e., it follows a Gaussian distribution! Idea: ICA therefore expects a Gaussian distributed input - which makes applying a second layer of ICA to a sparsely distributed input unlikely to work well. Hence, we apply a component-wise nonlinearity to the first layer outputs to make the output follow a Gaussian distribution.

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Recursive Independent Components Analysis (RICA 1.0) Another observation: The sign of the output of any s i is redundant statistically: This is the distribution of one signal source (s 2 ) as a function of the value of a neighboring signal source (s 1 ) …

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Recursive Independent Components Analysis (RICA 1.0) Hence our nonlinear activation function: Note that ambiguous (not quite on, not quite off) responses (in BLUE) are emphasized in the activation function

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Recursive Independent Components Analysis (RICA 1.0) An actual nonlinear activation function

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Recursive Independent Components Analysis (RICA 1.0) We applied RICA 1.0 to natural image patches. Layer-1 ICA learns the standard edge/bar shaped visual features. Layer-2 ICA learns more complex visual features that appear to capture contour and texture (Shan, Zhang & Cottrell, NIPS, 2007).

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Recursive Independent Components Analysis (RICA 1.0) Furthermore, these nonlinear features are useful: We applied layer 1 features with the nonlinearity to face recognition, and obtained state-of-the-art recognition performance on face recognition, using a simple linear classifier (Shan & Cottrell, CVPR, 2008). We also used the layer 1 features in a completely different recognition system we applied to faces, objects and flowers, and got state-of-the-art results on all three, without retraining (Kanan & Cottrell, CVPR, 2010)

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Results (NIPS 06/07) Error rates on the Yale face database: Number of training examples Number of features

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CVPR 2010 Both the salience map and the features stored at each location are ICA features with our nonlinearity Image Fixate Region Local Feature s Decide where to look Local Classifie r Decision? Get next fixation?

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Improvement Over State-of-the-art

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Roadmap We describe ICA and our version of a hierarchical ICA, Recursive ICA (RICA 1.0) We illustrate PCA and describe sparse PCA (SPCA) and the initial results We investigate the receptive fields of the higher layers of RICA 2.0.

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The “hidden layers” The story I told: Recursive Independent Components Analysis: ICA->add nonlinearity->ICA->add nonlinearity… The truth: in fact, like everyone else who does this sort of work, it is actually interleaved with PCA: PCA->ICA->add nonlinearity->PCA->ICA->add nonlinearity… And like everyone else, we never publish the pictures of the PCA receptive fields - because they don’t look biologically plausible!

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RICA 2.0 We now combine this with our improvements to sparse PCA (Vincent et al., 2005) to get receptive fields up to V2. SPCA->ICA->add nonlinearity->SPCA->ICA->add nonlinearity… And, sparse PCA learns biologically-realistic receptive fields.

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A simple (unrealistic) example Suppose two input signals (e.g., pixels) are completely correlated: Pixel 1 Pixel 2

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A simple (unrealistic) example Then we could represent that information with one (linear) “neuron”: This is an example of redundancy reduction Pixel 1 Pixel 2 0.5 0.5*Pixel 1 + 0.5*Pixel 2

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A simple (unrealistic) example Furthermore, we can reconstruct the original pixels from that one “neural response”: 0.5*Pixel 1 + 0.5*Pixel 2 Pixel 1 Pixel 2 1.0

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A simple (unrealistic) example Hence the “autoencoder network”: Pixel 1 Pixel 2 0.5 0.5*Pixel 1 + 0.5*Pixel 2 Pixel 1 Pixel 2 1.0

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Principal Components Analysis Principal Components Analysis would do exactly this, because it learns representations based on correlations between the inputs. This is an example of redundancy reduction and dimensionality reduction (from 2 dimensions to 1) Pixel 1 Pixel 2 0.5 0.5*Pixel 1 + 0.5*Pixel 2

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Principal Components Analysis Note that we can plot this “principal component” in image space, corresponding to the “weights”, (0.5,0.5) The same thing applies if we have more than two pixels…so we have more than 2 principal components…capturing more correlations… Pixel 1 Pixel 2 0.5 0.5*Pixel 1 + 0.5*Pixel 2 Pixel 1 Pixel 2

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Principal Components Analysis And now we can see that the reconstruction is a weighted version of that “image” The same thing applies if we have more than two pixels…so we have more than 2 principal components…capturing more correlations… 0.5*Pixel 1 + 0.5*Pixel 2 Pixel 1 Pixel 2 1.0 Pixel 1 Pixel 2

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Principal Components Analysis Here are the principal components of 10x10 patches of natural images:

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Principal Components Analysis But PCA learns these correlations in order of their size: so the first principal component does a lot of work: 1st PC

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Principal Components Analysis and the last principal component does very little work: last PC

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Principal Components Analysis So we can throw a lot of them away and you can’t tell the difference in an image that was reconstructed from them: Original Compressed So PCA does two things right: It decorrelates the inputs, and it reduces dimensionality, making it “efficient” at encoding images.

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Principal Components Analysis But no neuron should have to be the first principal component: So we should distribute the load evenly - this is called “response equalization.”

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Principal Components Analysis Secondly, PCA is profligate with connections - every pixel is connected to every principal component “neuron”: we should try to reduce the connections also.

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Sparse Principal Components Analysis We will try to minimize reconstruction error, While trying to equalize the neural responses And minimizing the connections.

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We minimize: Subject to the following constraint: Sparse Principal Components Analysis Reconstruction error Minimize connections Equalize the “work”

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Information Kept With Sparse Connections We applied the model to 20 X 20 image patches, and reduced the dimensionality to 100. Results: Our model captures 99.23% of the variance that could be captured by PCA with 100 output neurons. 96.31% of the connection weights in our model are zero.

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Information Kept With Sparse Connections

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The model as a neural net… It is A T that is mostly 0 …

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Results On grayscale images: Note that we get essentially the same results applying the model to pink noise images…

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Results suggesting the 1/f power spectrum of images is where this is coming from…

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Results On color images: Many people have gotten this color opponency before, but not in center-surround shape.

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Results The role of the number of features: 100 versus 32

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Results The role of : Recall this reduces the number of connections…

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Results The role of : higher means fewer connections, which alters the contrast sensitivity function (CSF). Matches recent data on malnourished kids and their CSF’s: lower sensitivity at low spatial frequencies, but slightly better at high than normal controls…

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Trained on grayscale video…

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Results Two “movies” (spatiotemporal features) laid out in time There appear to be two kinds of “cells.” TIM E Persistent, small Brief, big

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Midget? Parasol? This suggests that these cell types exist because they are useful for efficiently encoding the temporal dynamics of the world. Persistent, small Brief, big

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Roadmap We describe ICA and our version of a hierarchical ICA, Recursive ICA (RICA 1.0) We illustrate PCA and describe sparse PCA (SPCA) and the initial results We investigate the receptive fields of the higher layers of RICA 2.0.

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Recursive Independent Components Analysis (RICA 1.0) While in this talk, we only go as far as layer 2, obviously, we could keep going. Our goal is to check whether we are consistent with the neurophysiology before continuing.

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Enter Sparse PCA: RICA 2.0=RICA 1.0 + Sparse PCA Now, we no longer have to hide the PCA results! Question: What happens when we apply sparse PCA to the (nonlinearized) ICA outputs of the first layer??

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Image dataset Kyoto natural images… (Doi et al., 2003)

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Results Layer 1 Sparse PCA: Layer 1 ICA: (learned on 32x32 image patches)

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Results Layer 1 ICA (we used overcomplete ICA)

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Roadmap We describe ICA and our version of a hierarchical ICA, Recursive ICA (RICA 1.0) We illustrate PCA and describe sparse PCA (SPCA) and the initial results We investigate the receptive fields of the higher layers of RICA 2.0.

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How do neurophysiologists measure receptive fields of neurons?: Spike-triggered average, a.k.a. reverse correlation The reverse correlation method is used in neuroscience to characterize the properties of the retinal ganglion cells, the LGN cells, and the V1 simple cells. Neuroscientists present the animals with image patches of white Gaussian noise x, record the neuron’s response s. Then the receptive fields of the measured neuron is given by: If the neuron is a linear filter:

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Spike-triggered average on real cells…

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Visualization We take these: on top of one another And fit gabor filters to them, and then plot them on top of one another showing the major axis Then we color them by the strength of connections to them by the next layer - to show a receptive field.

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Layer 2 Sparse PCA cells Each patch represents one layer-2 PCA feature. Within the patch, each bar represents one layer-1 ICA feature. The layer-1 ICA features are fitted to Gabor kernel functions. The locations, orientations, lengths of the bars represent the locations, orientations, frequencies of the fitted Gabor functions. The colors of the bars represent the connection strengths from the layer-2 PCA feature to the layer-1 ICA features. Warm colors represent positive connections; Cold colors represent negative connections; Gray colors represent connection strengths that are close to zero.

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Layer 2 Sparse PCA cells A positive connection suggests that this layer-2 PCA features prefers strong responses, either positive or negative, from that layer-1 ICA feature. A negative connection suggests that it prefers weak or no responses from that layer-1 ICA feature. These perform the pooling operation on V1 simple cell responses: I.e., they agree with complex cell responses - but they also represent “OFF”-pooling responses (the cold colors)

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Layer 2 ICA features Unlike previous visual layers, there isn’t a general consensus about the visual features captured by V2 cells. We choose to compare our learned features with (Anzai, Peng, & Van Essen, Nature Neuroscience, 2007), because (1)It is a recent result, so they may reflect the most recent views about V2 cells from experimental neuroscience; (2)They adopted a similar technique as how we visualize the second layer features to describe the V2 cells’ receptive fields, hence it is convenient to compare our results with their results.

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Layer 2 ICA features They recorded 136 V2 cells from 16 macaque monkeys, but only reported the results on 118 of them (we will go back to this point later!!!) For each V2 cell, they first identified its classical receptive fields. Then they displayed 19 bars arranged in hexagonal arrays within the receptive field, whose sizes are much smaller than the receptive field size. They varied the orientations of the bars, and measured the V2 neurons’ responses to those settings. In the end, they got a space-orientation RF map for each V2 neuron.

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Layer 2 ICA features In the end, they got a space-orientation RF map for each V2 neuron. The first shows uniform orientation tuning across its receptive field; the second, non-uniform tuning in different sub-regions of space.

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Layer 2 ICA features The left-most column displays two model neurons that show uniform orientation preference to layer-1 ICA features. The middle column displays model neurons that have non- uniform/varying orientation preference to layer-1 ICA features. The right column displays two model neurons that have location preference, but no orientation preference, to layer-1 ICA features.

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Layer 2 ICA features The first two columns are consistent with their findings. The right column does not, but they “threw away” 18 of their cells that did not show orientation tuning.

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Summary Dimensionality Reduction (e.g., Sparse PCA) & Expansion (e.g., overcomplete ICA) might be a general strategy of information processing in the brain. The first step removes noise and reduces complexity, the second step captures the statistical structure. We showed that retinal ganglion cells and V1 complex cells may be derived from the same learning algorithm, applied to pixels in one case, and V1 simple cell outputs in the second. This highly simplified model of early vision is the first one that learns the RFs of all early visual layers, using a consistent theory - the efficient coding theory. We believe it could serve as a basis for more sophisticated models of early vision.

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END

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