1 Introduction
According to prevailing models, the mammalian visual cortex is organized in a hierarchy of levels that encode increasingly higher levels features from edges to object categories. The primary visual area V1 contains simple cells, which primarily respond to oriented edges at particular locations, orientations and frequencies, and complex cells which appear to pool the outputs of multiple simple cells over a range of locations. Receptive fields similar to simple cells have been shown to be produced by sparse coding algorithms (Olshausen and Field, 1996).
The precise computation carried out by complex cells is not entirely elucidated. One idea is that they pool simple cells that often respond consecutively in time. In the socalled slow feature analysis method (Wiskott and Sejnowski, 2002; Berkes and Wiskott, 2005; Bergstra and Bengio, 2009), this is achieved by penalizing the rate of change of unit activations as the input varies. Another approach is to impose sparsity constraints on complex cells that pool local groups of simple cell (Hyvarinen and Hoyer, 2001; Kavukcuoglu et al., 2009). This forces units within a local group to learn similar filters that are often coactivated. Another approach, applied to static image patches, is to model the covariance of images, increasing the likelihood of features that commonly occur together, forming their representation (Karklin and Lewicki, 2008). In Cadieu and Olshausen (2009) the image is represented as a sparse model in terms amplitude and phase at the first layer and the second layer discovers translational invariants. Another structured model of video is Berkes et al. (2009).
This paper first demonstrates that sparse coding can be used to train a network of locallyconnected simple cells that operates on an image of arbitrary size. The network is composed of a feedforward encoder which computes an approximation to the sparse features, and a decoder which reconstructs the input from the sparse features (Kavukcuoglu et al., 2008, 2009). Unlike models such convolutional networks (LeCun et al., 1998), the filters are not shared across locations. The absence of shared weights is more biologically plausible than convolutional models that assume identical replicas of filters over the input field. The second section of the paper introduces the use of sparsity criteria operating on local pools of simple cells (Hyvarinen and Hoyer, 2001; Kavukcuoglu et al., 2009). Since the sparsity drives the number of active blocks to be small, simple cells arrange themselves so that similar filter (which often fire together) group themselves within pools. In a large, locally connected network, this will result in orientationselective, simplecell filters that are organized in pinwheel patterns, similar to those observed in the primate’s visual cortex. The third section introduces the temporal product network that is designed to discover representations that are invariant over multiple consecutive inputs. This produces units that respond to edges of a given orientation and frequency but over a wide range of positions, similar to the complex cells of V1. The model includes a feedforward encoder architecture that can produce the internal representation through a simple feedforward propagation, without requiring an optimization process.
The architecture described below is applicable to any slowlyvarying sequential signal, but we will focus the discussion on temporal sequences of images (video).
2 Sparse Feature Learning in a LocallyConnected Network
Predictive Sparse Decomposition (PSD) (Kavukcuoglu et al., 2008, 2009)
is based on Olshausen and Field’s sparse coding algorithm in which a decoding matrix is trained so that image patches can be reconstructed linearly by multiplying the decoding matrix by a sparse feature vector
(Olshausen and Field, 1996). Unlike with sparse coding, PSD contains an efficient feedforward encoder (a nonlinear regressor), which is trained to map input images to approximate sparse feature vectors from which the input can be linearly reconstructed.As with sparse coding, PSD is normally trained on individual image patches. Applying the resulting filters to a large image results in redundant representations, because the learning algorithm contains no mechanism to prevent a high degree of redundancy between outputs of the same filter (or similar filters) at neighboring locations. In this section, we introduce a form of PSD that is applied to locallyconnected networks of units whose receptive fields are uniformly spread over a large image.
2.1 Sparse coding with an encoder.
The basis of the PSD algorithm is Olshausen and Field’s sparse coding method for learning overcomplete basis functions (Olshausen and Field, 1996). We denote by the input vector (an image patch) of dimension and by a (sparse) feature vector of dimension from which the input is reconstructed. The reconstructed input is produced through a linear decoder , where is an decoding matrix (or dictionary matrix) to be learned, whose columns have norm 1 and are interpreted as basis vectors. Given a decoding matrix , sparse coding inference consists in finding the feature vector that minimizes the energy function
(1) 
where denotes the norm of (sum of absolute values of the components). The positive constant controls the sparsity penalty.
(2) 
The learning algorithm uses a gradientbased method to find the matrix that minimizes the average of the following energy function over a training set of input vectors
(3) 
In PSD, a parameterized encoder function is trained to compute a prediction of the optimal sparse vector . In its simplest form, the encoder takes the form
(4) 
where collectively denotes the encoding matrix , the diagonal matrix , and the
dimensional bias vector
. In PSD, the encoder and decoder are trained simultaneously. The optimal code minimizes the following energy function(5) 
As with Sparse Coding, the PSD training procedure uses a gradientbased method to find the and that minimize the following objective function averaged over a training set of input vectors
(6) 
An iteration of the training procedure is as follows. Given an input vector and the current parameters, compute . Then initialize , and find the that minimizes , using gradient descent or some other iterative method. Update the parameters , , , and so as to lower
, using a step of stochastic gradient descent. finally, renormalize the columns of
to unit norm. The normalization of is necessary to prevent singular solutions in which is very small, and very large. After training on natural image patches, the columns of and the rows of become oriented edge detectors. The inferred for a typical image patch will be sparse, and the predicted will be quite close to the optimal for any near the manifold of high training samples density. The encoder function provides a very efficient (feedforward) way to produce an approximation of the optimal . We interpret the rows of as filters (or receptive fields) and the components of as simple cell activations.2.2 Locallyconnected network.
While the original PSD method is trained on individual patches, our aim is to train an entire set of local filters over a large image using PSD. We must point out that filters with different receptive fields are not constrained to be identical. This is very much unlike “convolutional” approaches in which the weights of filters at different locations are shared. Basically, a given simple cell (a given component of ) is connected only to a local receptive field in the input image. Similarly, the corresponding component in is connected to the same “projection field” in the input through the decoder. In general the receptive fields can have arbitrary shapes, not all inputs need to be connected, and different location can have different densities of simple cells (e.g. a density that geometrically decreases with excentricity, as in the primates’ visual systems). In the simplest case used here, the connectivity is uniform: the simple cells form a twodimensional regular grid over the image. Each one is connected to a square receptive field directly below it. The density of simple cells can be set to be higher than that of the pixels. This produces an overcomplete representation in which several simple cells have the same receptive field (but different weights). The density of simple cells can also be lower than that of the pixels, corresponding to an undercomplete representation in which adjacent receptive fields are stepped by more than 1 pixel. The densities can be identical, producing a onetoone representation.
Formally, if , are the integer coordinates of a simple cell, then its receptive field has coordinates where , , where is the size of the neighborhood, are the densities of simple cells in the two directions, , is the image size and .
This network is considerably smaller than a fully connected network with the same size input. The number of connections goes from to where the image is of size and the local neighborhood of size , and is the overcompleteness factor. This makes training tractable for large images. Arguably, constraining the receptive fields to be local hardly reduces the capacity of the system, since sparse coding algorithm end up learning highly localized filters, and zeroing out most of the weights.
2.1 Periodic replication.
While our locallyconnected network can be trained on images of arbitrary sizes, there is little advantage to training it on images that are larger than a small multiple of the receptive field size. This is because the activations of simple cells that are away from each other are essentially independent of each other. Conversely, the activations of nearby simple cells depend on each other through the minimization under sparsity: neighboring units compete to explain the input, implementing a kind of “explaining away” mechanism. Hence we replicate a “tile” of weights in periodic fashion over the image. Other way to say this is that, in a locally connected network, we share (tie) those weights together that are multiple of an integer distance away from each in each direction. This allows us to train on smaller size inputs, such as pixels and apply it to an arbitrarily large image. This sharing takes advantage of the fact that the statistics of the image is the same at different points. If this periodicity is the same as the local neighborhood, the number of weights becomes  the same as that of the corresponding image patch. If the periodicity is 1, the system reduces to a convolutional layer.
Formally, let be the weight matrix element between simple cell at and pixel . Then where and are integers. Numbers are for overcomplete, for undercomplete and for complete system (in particular direction) where are integers, and the are also required to be integers. Note that for
the network reduces to convolutional neural network with
number of feature maps.2.2 Boundary Effects.
Units at the periphery of the network receive less than inputs, hence must be treated differently from regular units. If the image size on which the system is trained were very large, the effect of these units on the training process would be negligible. But it is more efficient and convenient to train on images that are as small as possible, generally around . Hence to avoid a adverse effects of the boundary units on learning, their weights are not shared with other units. With this method, there is no visible artifacts on the weights of the bulk units when training on images of size or greater.
2.3 Input Data and Preprocessing
The method was tested with two datasets. In the first, pixel windows were extracted from the Berkeley image dataset. Consecutive frames were produced by shifting the window over the original image by 1 or 2 pixels in any direction. For the second set of input, short sequences of consecutive frames were extracted from the movie “A Beautiful Mind”. Results are reported for the first dataset, but the results obtained with the second one were very similar.
Before extracting the windows and feeding then to the network, each image is preprocessed by removing the local mean (using a high pass filter) and normalizing the standard deviation (contrast normalization) as follows. First each pixel is replaced by its own value minus a gaussian weighted average of its neighbors. Then the pixel is divided by the gaussianweighted standard deviation of its neighbors. The width of both gaussians was
pixels. In the contrast normalization there was a smooth cutoff that rescales pixels with small standard deviation less.2.1 Training.
The training proceeds the same way as in the patchbased version of Kavukcuoglu et al. (2008). The optimal code is found using gradient descent with a fixed number of steps. The code inference takes more computation than with patchlevel training. To minimize the “batching effect” due to weight sharing, the weights are updated on the basis of the gradients contributed by units with a single common receptive field. After this update, the optimal code is adjusted with a small number of gradient descent iterations, and the process is repeated for the next receptive field. This procedure accelerates the training by making it more “stochastic” than if the weights were updated using the gradient contributions accumulated over the entire image.
2.2 Results.
Learned filters are shown in the Figure 1. As expected, oriented edge detectors are obtained, similar to those obtained by training on patches. There is one significant difference: in patchbased training, the filters have to cover all possible locations of edges within the patch. By contrast, our system can choose to use a unit with a neighboring receptive field to detect a shifted edge, rather than covering all possible locations within a single receptive field location. Hence, the units tend to cover the space of location, orientations and frequencies in a considerably more uniform fashion than if we simply replicate a system trained at the patch level. Most of the filters are in fact centered within their receptive field.
2.3 A better encoder.
The nonlinearity used in the original PSD encoder is of the form where , and is a diagonal gain matrix. Unfortunately, this encoder makes it very difficult for the system to produce sparse output, since a zero output is in the highgain region of the function. To produce sparse outputs, the nonlinearity would need a “notch” around zero, so that small filter responses will be mapped to zero. Our solution is to use a “double tanh” function of the form where is a learned parameter that determines the width of the “notch”. The prediction error is empirically better by about factor of two for complete network with this doubletanh than with the regular tanh.
3 Pinwheel patterns through group sparsity.
Hubel and Wiesel’s classic work showed that oriented edge detectors in V1 are organized in topographic maps such that neighboring cells respond to similar orientations or frequencies at nearby locations. Local groups of units can be pooled by complex cells with responses that are invariant to small transformation of the inputs. Hyvarinen and Hoyer have proposed to use group sparsity constraints in a sparse reconstruction framework to force similar filters to gather within groups (Hyvarinen and Hoyer, 2001). The outputs of units in a group are pooled by complex cells. Kavukcuoglu et al. have proposed a modification of the PSD method that uses this idea to produce invariant complex cells (Kavukcuoglu et al., 2009). Here, we propose to use the same idea to produce topographic maps over real space: filters that are nearby in real space will also detect similar features. The new sparsity criterion for a single pool is:
(7) 
where is the vector of the coordinates of a simple cell and is an integer vector. The overall criterion is the sum of these over the entire domain (the pools overlap). This term tends to minimize the number of pools that have active units, but does not prevent multiple units from being simultaneously active within a pool. Hence pools tend to regroup filters that tend to fire together.
We can apply this to a locally connected network in a natural way, as the simple cells are already distributed on a two dimensional grid. The result for a periodic locally connected network with local neighborhood of size , periodicity , overcomplete is shown in the Figure 2a. We see that the network puts the filters of the similar orientation and frequency close to each other. Due to the topology of putting orientations in the periodic grid, it is impossible to have smooth transitions everywhere, which results in point topological defects  and pinwheels patterns around them, familiar to neurophysiologists. These are clearly visible in the Figure 2a which are marked by red circles/line.
In the periodic network, these have to fit periodically into the square grid (on a torus). There is no periodicity in the brain, and one has the usual maps over the whole area of V1 (Obermayer and Blasdel, 1993; Crair et al., 1997) with pinwheels distributed in somewhat random nonperiodic fashion (on a randomly deformed grid). This is easily implemented here, in the locally connected network without periodicity. We took the input of size with local neighborhoods of sizes and a complete case (). At the location of each filter we draw a pixel with a color whose hue is proportional to the orientation. This is shown in the Figure 2b. It very much resembles the maps obtained from the monkey cortex in the reference (Obermayer and Blasdel, 1993).
4 Temporal product network for Invariant Representations
While the elementary feature detection performed by simple cells is a good first step, perception tasks require higherlevel features with invariance properties. In the standard model of early vision, complex cells pool similar features in a local neighborhood to produce locally invariant features. The process eliminates some information.
The present section introduces an alternative method to learn complex cells that are invariant to small transformations of the inputs. The method preserves all the information in the image by separately encoding the “what” and the “where” information. As with slow feature analysis (Wiskott and Sejnowski, 2002), the main idea is to exploit temporal constancy in video sequences. The system is again built around the encoderdecoder concept, but the encoder and the decoder are quite different from PSD’s. The key ingredient is multiplicative interactions. Hinton and his collaborators have proposed several temporal models that use 3way multiplicative interactions (Brown and Hinton, 2001; Taylor and Hinton, 2009). Our model is different in that two state vector of identical sizes are multiplied term by term.
The product network described here operates on the output of simple cells described in the previous sections, trained beforehand. More precisely, the input to the product network are the absolute values of the simple cell activations produced by the encoder discussed above. The results are given with simple cells trained with “simple” sparsity, but the results obtained with the simple cells trained with group sparsity are qualitatively similar.
4.1 Separating the “What” from the “Where”.
The basic idea is to split the complex cells into two complementary groups: invariant cells and location cells. The state of the invariant cells is constrained to be constant for several consecutive frames from a video. These cells will encode the content of the frames, regardless of the position at which the content appears in the frame. The complementary location cells encode the location of the content, which may change between frames. The two codes cooperate to reconstruct the input.
As an example, let us consider each input to be an edge at a particular orientation that moves over time. Different simple cells (at different locations) would respond to each frame. For simplicity we can imagine that there is one simple cell active for each edge/frame, though in reality the representation, while sparse, has multiple active cells. After training, each invariant complex cell would respond to edges of a particular orientation at several positions. During reconstruction, it would reconstruct all these at every frame (the values of the simple cells corresponding to these edges). Each complementary (position) cell would respond to edges at a certain position but of various orientations. Different complementary cells would be active at different time frames. At a given frame, an active complementary cell would reconstruct the input edge at that frame along with edges of other orientations at that position that it is connected to. Taking the product of the reconstructions coming from the invariant cells and the complementary cells gives the desired input edge. Thus the input is translated into a more useful representation: the orientation and the location.
4.2 Encoder and decoder architectures.
The detailed architecture of the decoder is given in figure 3. Let the input to the temporal product network at time be (the values of the simple cells). At this time we consider frames  the current one and the consecutive previous ones. Let the invariant code be denoted by . At this there is one complementary code for each time frame, denoted by where . The invariant code tries to turn on all the related simple cells at these time frames, and the complementary code selects the correct one at each time frame. The reconstructed input (the decoder operation) for the at time is
(8) 
Here the are matrices, the dot denotes the matrix multiplication and the cross the term by term multiplication of the vectors. The columns of , are normalized and the , are nonnegative. The energy to be minimized is
(9) 
where we typically have .
The form of equations (8,9) is not arbitrary. In this paragraph we give three intuitive arguments from which this form follows. The first two arguments are the same as for the simple cell network. First, the normalization of the columns  sum of the squares equals to one  relative to the power of sparsity: one  is what causes sparse representations to have a lower energy. There could be different powers but the normalization power needs to be greater then the sparsity power. Second the sparsity power should be one for the following reason. Imagine the power was larger then one and we have two filters which are similar. Then given an input that perfectly matches the first filter, the other filter would also turn on because with power greater then one it is advantageous to distribute activity among both. Furthermore, this would pull the filters together. On the other hand if power was smaller then one, a given input would tend to commit to one of the units even though the other would also be a good explanation, though this might be acceptable. Third, there should be square root for the following reason. Imagine there wasn’t and that we have an input that can be well reconstructed. If we start with a small code, the gradient from the first term would be small (proportional to the code) but the gradient from the second term would be constant. Thus we would end up with a zero code even though there is a perfectly good code that reconstructs the input. What we need is that the size of the gradient is independent of the magnitude of the starting code (assume it is nonzero). Square root has this property. We have done some experiments without the square root and obtained invariant filters as well, but their diversity is not as good as of those obtained with the square root, especially in the locally connected network.
The encoder module is defined as follows:
(10)  
(11) 
where ’s are the encoder matrices, ,,, are vectors and , are scalars.
4.3 Comparison to slow feature analysis.
In temporal product network, at every times step we are inferring a code for several time frames. When we move by one time step, new set of codes will be inferred. This is different from the slow feature analysis. There a problem is that after arbitrary number of steps you get artifacts from the previous times. Here code is inferred only on the fixed set of frames. In a transition period between two invariant features the network might not be able to reconstruct the input properly, but as soon as we are well into the new feature, the reconstruction doesn’t have any artifacts. In fact, it might be possible, but we haven’t tested it, that even in the transitional period, the reconstruction works well as follows. Invariant units for both invariant features would be on, but if the complementary connections don’t have overlaps, the reconstruction is good.
5 Results.
To help understand the function of the network, we show results for a toy example in figure 4. The input to the temporal product network is a image patch whose values are given by a gaussian of width pixels at a location . The gaussian moves to the right and when it disappears on the right, it appears on the left at a randomly generated . We see that the cells are invariant to the position (direction of movement) in some range of and are therefore invariant to shifts that happen in time. The cells are complementary and are invariant to the range of values . In terms of direction of motion they extract position. More precisely their goal is to group inputs that are similar but are not along the direction of motion.
Now we discuss the realistic example of image training. We start with the discussion of image patch training. The results for locally connected network are below. The Figure 5b shows a selection of the simple cell filters. We fitted each of the simple cell filters with a Gabor function, giving us among other parameters, orientation, position and frequency of the filter. The Figure 5d shows a selection of complex cells. Each line correspond to a simple cell filter, with orientation and position obtained from the fit, and the intensity proportional to the strength of the connection between the complex and simple cells. We see that each invariant cell has strong connections to edge detectors of similar orientation at a range of positions.
Next, we look at the responses of the simple and invariant cells to moving edges. We parameterize an edge by orientation and position, the later being the distance from the center of the input patch. The responses of the simple cells are in the Figure 5a. Each color correspond to a different simple cell and the size of the bubble is proportional to the activity. Analogous graph for the invariant cells is shown in the Figure 5c. We see that the invariant cells respond to much larger range of positions then the simple cells but a similar range of orientations. We also see that the responses are quite smooth. (The edge was moving very slowly, practically stationary. For faster moving edge the responses are even smoother.)
Next we discuss the diversity of the filters. For the system to perform well it should have filters distributed evenly among orientations and have a diversity of frequencies. This time we show the results for the locally connected network. The simple cell network used here has a local neighborhood of size , is overcomplete and with periodicity in the space of in each direction. The complex cell network has local neighborhood of size , undercomplete with respect to the simple cell layer, with periodicity in each direction. The orientation/frequency plot for the simple cells is in the Figure 6a and for the complex cells in the Figure 6b. The radius is the frequency and the orientation is twice the angle from the axis. For the complex cell there is no such number, but since they have strong connections to edges of similar orientation and frequency we calculate the average weighted by the square of this connection. We see that in both cases we have a smooth distribution in the frequency/orientation space. This is where the form (8) of the reconstruction is important. Without the square root for example the parameters are not so well distributed.
The final system that we obtain can be applied to large images and used for fast image recognition as it contains a feedforward pass through the whole system. Let us recapitulate the computations involved in the forward pass calculation of activities of complex cells. The preprocessing contains two convolutions  for mean and standard deviation removal. The simple cell calculation is not a convolution but application of different filter at different point (followed by nonlinearity). However, the computational cost is equivalent to computing a number of convolutions equal to the overcompleteness of the system. Results are presented for a complete (but not overcomplete) system, hence the cost is equal to a single convolution. The next level, which contains the complex cells, also involves the application of different filters at each point, followed by a nonlinearity. In this case we use four times undercomplete system and hence the computational cost is equivalent to one quarter of a convolution. Afterwards we train logistic regression classifier.
6 Efficiency of locally connected organization
Convolutional net is a special case of a periodic locally connected net when the periodicity in the input space is one in each direction. This arrangement appears sensible since the input statistics is translationaly invariant. However if we have a limited computational capacity, there is only a limited number of filters in the image that we can use. It is likely that for a given filter (especially a low frequency one) it is enough to apply it with spacing of two or more pixels. This frees up additional resources to use for filters of different type and allows richer model. However the question is: What is the right spacing to use? It is likely that large spacing is enough for low frequency filters so the spacing is ideally filter dependent. However filters are elongated along one direction, so the ideal spacing is likely to be orientation dependent as well. Ideally the network should discover the correct allocation of filters on its own. This is precisely what locally connected network does. However with the locally connected network, we have lost translational invariance, for example we don’t know which filters to pool together in the pooling layer of convolutional net. Hence we need to learn this pooling. This is what the temporal product network does. Further it can learn to pool appropriate slightly different orientations together since they are all activated for a given edge.
To test these ideas we trained locally connected network unsupervised on Berkeley images as described above. We used filters and complete simple cell layer (density of simple cells equals density of inputs). The number of computations of the feedforward pass is the same as that of one convolution. For the complex cell layer we used filters and four times undercomplete system. The number of computations this time is that of a quarter of convolutions. Note that preprocessing contained two convolutions. We testing the performance on Caltech 101 dataset (FeiFei et al., 2007) with 30 training images per category. The resulting performance was . Performing a local subtraction and contrast normalization on the top layer improves it to .
These results are not state of the art which is currently for systems of this class is Boureau et al. (2010) (the system extracts sift features, then learns sparse dictionaries, pools and makes histograms). There are several system of this class (Lazebnik et al., 2006; Pinto et al., 2008, 2009; Serre et al., 2007; Yang et al., 2009; Lee et al., 2009; Yu et al., 2009) but specifically convolutional nets achieve (Jarrett et al., 2009). However this system should be compared to a single layer convolutional net, since it essentially consists of filters with nonlinearity (simple cells) and pooling (complex cells). Single layer neural network achieves about performance which is the same as this network. However convolutional net is much larger, it typically involves convolutions. Thus the locally connected net can achieve the same performance at lower computational cost. This gives the merit to the idea descried above that locally connected organization is more efficient then convolutional one. However more experimental evaluation is needed.
7 Conclusion
We have presented a new neural architecture that follows more closely the kind of calculations performed by the visual cortex but which at the same time can be used for real time object recognition. It is a layered architecture. It’s first layer is a locally connected version of PSD architecture. It’s main feature is that the weights are not shared for nearby filters (but can be for filters at larger distances for efficiency) and the geometry is smooth, e.i. contains no cuts. The next layer features a new algorithm for invariance extraction from temporal data. It’s aim is to translate the input into two types of information  the “what” information that is invariant and the “where” information that complements the invariant one. This layer is also designed in a locally connected way. Both layers include encoder that predicts the values of the cells in a fast feedforward fashion. Therefore by including one of the standard classifiers (logistic regression in out case) the whole system can be used for fast visual recognition. As the system is smaller then other ones typically used, the recognition is faster but the performance is lower. It is left for the future work to see how this performance can be improved.
This architecture suggests that locally connected organization without sharing of nearby weights is more efficient the convolutional one because it allocates correct filters at every location, rather then applying the same filter unnecessarily often.
In the future we need to increase the performance of the system, use more overcomplete representations, train more layers of the system and show conclusively, if true, that locally connected training is more efficient the convolutional one.
Acknowledgments
We thank Koray Kavukcuoglu for useful discussions.
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