CBMM Memo Series
https://hdl.handle.net/1721.1/88531
2019-09-23T04:15:46ZHippocampal Remapping as Hidden State Inference
https://hdl.handle.net/1721.1/122040
Hippocampal Remapping as Hidden State Inference
Sanders, Honi; Wilson, Matthew A.; Gershman, Samueal J.
Cells in the hippocampus tuned to spatial location (place cells) typically change their tuning when an animal changes context, a phenomenon known as remapping. A fundamental challenge to understanding remapping is the fact that what counts as a “context change” has never been precisely defined. Furthermore, different remapping phenomena have been classified on the basis of how much the tuning changes after different types and degrees of context change, but the relationship between these variables is not clear. We address these ambiguities by formalizing remapping in terms of hidden state inference. According to this view, remapping does not directly reflect objective, observable properties of the environment, but rather subjective beliefs about the hidden state of the environment. We show how the hidden state framework can resolve a number of puzzles about the nature of remapping.
2019-08-22T00:00:00ZBrain Signals Localization by Alternating Projections
https://hdl.handle.net/1721.1/122034
Brain Signals Localization by Alternating Projections
Adler, Amir; Wax, Mati; Pantazis, Dimitrios
We present a novel solution to the problem of localization of brain signals. The solution is sequential and iterative, and is based on minimizing the least-squares (LS) criterion by the alternating projection (AP) algorithm, well known in the context of array signal processing. Unlike existing solutions belonging to the linearly constrained minimum variance (LCMV) and to the multiple-signal classification (MUSIC) families, the algorithm is applicable even in the case of a single sample and in the case of synchronous sources. The performance of the solution is demonstrated via simulations.
2019-08-29T00:00:00ZTheoretical Issues in Deep Networks: Approximation, Optimization and Generalization
https://hdl.handle.net/1721.1/122014
Theoretical Issues in Deep Networks: Approximation, Optimization and Generalization
Poggio, Tomaso; Banburski, Andrzej; Liao, Qianli
While deep learning is successful in a number of applications, it is not yet well understood theoretically. A satisfactory theoretical characterization of deep learning however, is beginning to emerge. It covers the following questions: 1) representation power of deep networks 2) optimization of the empirical risk 3) generalization properties of gradient descent techniques - why the expected error does not suffer, despite the absence of explicit regularization, when the networks are overparametrized? In this review we discuss recent advances in the three areas. In approximation theory both shallow and deep networks have been shown to approximate any continuous functions on a bounded domain at the expense of an exponential number of parameters (exponential in the dimensionality of the function). However, for a subset of compositional functions, deep networks of the convolutional type (even without weight sharing) can have a linear dependence on dimensionality, unlike shallow networks. In optimization we discuss the loss landscape for the exponential loss function. It turns out that global minima at infinity are completely degenerate. The other critical points of the gradient are less degenerate, with at least one - and typically more - nonzero eigenvalues. This suggests that stochastic gradient descent will find with high probability the global minima. To address the question of generalization for classification tasks, we use classical uniform convergence results to justify minimizing a surrogate exponential-type loss function under a unit norm constraint on the weight matrix at each layer. It is an interesting side remark, that such minimization for (homogeneous) ReLU deep networks implies maximization of the margin. The resulting constrained gradient system turns out to be identical to the well-known {\it weight normalization} technique, originally motivated from a rather different way. We also show that standard gradient descent contains an implicit L2 unit norm constraint in the sense that it solves the same constrained minimization problem with the same critical points (but a different dynamics). Our approach, which is supported by several independent new results, offers a solution to the puzzle about generalization performance of deep overparametrized ReLU networks, uncovering the origin of the underlying hidden complexity control in the case of deep networks.
2019-08-17T00:00:00ZFunction approximation by deep networks
https://hdl.handle.net/1721.1/121183
Function approximation by deep networks
Mhaskar, H.N.; Poggio, Tomaso
We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same compositional structure, while a shallow network cannot exploit this knowledge. Thus, the blessing of compositionality mitigates the curse of dimensionality. On the other hand, a theorem called good propagation of errors allows to “lift” theorems about shallow networks to those about deep networks with an appropriate choice of norms, smoothness, etc. We illustrate this in three contexts where each channel in the deep network calculates a spherical polynomial, a non-smooth ReLU network, or another zonal function network related closely with the ReLU network.
2019-05-30T00:00:00Z