Neural Computation最新文献

The visual system performs a remarkable feat: it takes complex retinal activation patterns and decodes them for object recognition. This operation, termed "representational untangling," organizes neural representations by clustering similar objects together while separating different categories of objects. While representational untangling is usually associated with higher-order visual areas like the inferior temporal cortex, it remains unclear how the early visual system contributes to this process-whether through highly selective neurons or high-dimensional population codes. This article investigates how a computational model of early vision contributes to representational untangling. Using a computational visual hierarchy and two different data sets consisting of numerals and objects, we demonstrate that simulated complex cells significantly contribute to representational untangling for object recognition. Our findings challenge prior theories by showing that untangling does not depend on skewed, sparse, or high-dimensional representations. Instead, simulated complex cells reformat visual information into a low-dimensional, yet more separable, neural code, striking a balance between representational untangling and computational efficiency.

When applying nonnegative matrix factorization (NMF), the rank parameter is generally unknown. This rank, called the nonnegative rank, is usually estimated heuristically since computing its exact value is NP-hard. In this work, we propose an approximation method to estimate the rank on the fly while solving NMF. We use the sum-of-norm (SON), a group-lasso structure that encourages pairwise similarity, to reduce the rank of a factor matrix when the initial rank is overestimated. On various data sets, SON-NMF can reveal the correct nonnegative rank of the data without prior knowledge or parameter tuning. SON-NMF is a nonconvex, nonsmooth, nonseparable, and nonproximable problem, making it nontrivial to solve. First, since rank estimation in NMF is NP-hard, the proposed approach does not benefit from lower computational complexity. Using a graph-theoretic argument, we prove that the complexity of SON NMF is essentially irreducible. Second, the per iteration cost of algorithms for SON-NMF can be high. This motivates us to propose a first-order BCD algorithm that approximately solves SON-NMF with low per iteration cost via the proximal average operator. SON-NMF exhibits favorable features for applications. Besides the ability to automatically estimate the rank from data, SON-NMF can handle rank-deficient data matrices and detect weak components with little energy. Furthermore, in hyperspectral imaging, SON-NMF naturally addresses the issue of spectral variability.

Operator learning is a recent development in the simulation of partial differential equations by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network is an approximate mapping in infinite-dimensional spaces that is capable of (approximately) simulating the solution operator governed by the partial differential equation. In our work, we study some general approximation capabilities for linear differential operators by approximating the corresponding symbol in the Fourier domain. Analogous to the structure of the class of Hörmander symbols, we consider the approximation with respect to a topology that is induced by a sequence of semi-norms. In that sense, we measure the approximation error in terms of a Fréchet metric, and our main result identifies sufficient conditions for achieving a predefined approximation error. We then focus on a natural extension of our main theorem, in which we reduce the assumptions on the sequence of seminorms. Based on existing approximation results for the exponential spectral Barron space, we then present a concrete example of symbols that can be approximated well.