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The high dimensionality of hyperspectral images (HSI) that contains more than hundred bands (images) for the same region called Ground Truth Map, often imposes a heavy computational burden for image processing and complicates the learning process. In fact, the removal of irrelevant, noisy and redundant bands helps increase the classification accuracy. Band selection filter based on "Mutual Information" is a common technique for dimensionality reduction. In this paper, a categorization of dimensionality reduction methods according to the evaluation process is presented. Moreover, a new filter approach based on three variables mutual information is developed in order to measure band correlation for classification, it considers not only bands relevance but also bands interaction. The proposed approach is compared to a reproduced filter algorithm based on mutual information. Experimental results on HSI AVIRIS 92AV3C have shown that the proposed approach is very competitive, effective and outperforms the reproduced filter strategy performance. Keywords - Hyperspectral images, Classification, band Selection, Three variables Mutual Information, information gain.

The propagation delay is non-negligible in underwater acoustic networks (UANs) since the propagation speed is five orders of magnitude smaller than the speed of light. In this case, space and time factors are strongly coupled to determine the collisions of packet transmissions. To this end, this paper analyzes the impact of spatial-time coupling on slotted medium access control (MAC). We find that both inter-slot and intra-slot collisions may exist, and the inter-slot collision may span multiple slots. The sending slot dependent interference regions could be an annulus inside the whole transmission range. It is pointed out that there exist collision-free regions when a guard interval larger than a packet duration is used in the slot setting. In this sense, the long slot brings spatial reuse in a transmission range. However, we further find that the successful transmission probabilities and throughput are the same for the slot lengths of one packet duration and two packet durations. Simulation results show that the maximum successful transmission probability and throughput can be achieved by a guard interval less than a packet duration, which is much shorter than the existing slot setting in typical Slotted-ALOHA. Simulations also show that the spatial impact is greater for vertical transmission than for horizontal transmissions due to the longer vertical transmission range in three-dimensional UANs.

We present a scalable strategy for development of mesh-free hybrid neuro-symbolic partial differential equation solvers based on existing mesh-based numerical discretization methods. Particularly, this strategy can be used to efficiently train neural network surrogate models for the solution functions and operators of partial differential equations while retaining the accuracy and convergence properties of the state-of-the-art numerical solvers. The presented neural bootstrapping method (hereby dubbed NBM) is based on evaluation of the finite discretization residuals of the PDE system obtained on implicit Cartesian cells centered on a set of random collocation points with respect to trainable parameters of the neural network. We apply NBM to the important class of elliptic problems with jump conditions across irregular interfaces in three spatial dimensions. We show the method is convergent such that model accuracy improves by increasing number of collocation points in the domain. The algorithms presented here are implemented and released in a software package named JAX-DIPS (//github.com/JAX-DIPS/JAX-DIPS), standing for differentiable interfacial PDE solver. JAX-DIPS is purely developed in JAX, offering end-to-end differentiability from mesh generation to the higher level discretization abstractions, geometric integrations, and interpolations, thus facilitating research into use of differentiable algorithms for developing hybrid PDE solvers.

During the last decade, hyperspectral images have attracted increasing interest from researchers worldwide. They provide more detailed information about an observed area and allow an accurate target detection and precise discrimination of objects compared to classical RGB and multispectral images. Despite the great potentialities of hyperspectral technology, the analysis and exploitation of the large volume data remain a challenging task. The existence of irrelevant redundant and noisy images decreases the classification accuracy. As a result, dimensionality reduction is a mandatory step in order to select a minimal and effective images subset. In this paper, a new filter approach normalized mutual synergy (NMS) is proposed in order to detect relevant bands that are complementary in the class prediction better than the original hyperspectral cube data. The algorithm consists of two steps: images selection through normalized synergy information and pixel classification. The proposed approach measures the discriminative power of the selected bands based on a combination of their maximal normalized synergic information, minimum redundancy and maximal mutual information with the ground truth. A comparative study using the support vector machine (SVM) and k-nearest neighbor (KNN) classifiers is conducted to evaluate the proposed approach compared to the state of art band selection methods. Experimental results on three benchmark hyperspectral images proposed by the NASA "Aviris Indiana Pine", "Salinas" and "Pavia University" demonstrated the robustness, effectiveness and the discriminative power of the proposed approach over the literature approaches. Keywords: Hyperspectral images; target detection; pixel classification; dimensionality reduction; band selection; information theory; mutual information; normalized synergy

Recent mean field interpretations of learning dynamics in over-parameterized neural networks offer theoretical insights on the empirical success of first order optimization algorithms in finding global minima of the nonconvex risk landscape. In this paper, we explore applying mean field learning dynamics as a computational algorithm, rather than as an analytical tool. Specifically, we design a Sinkhorn regularized proximal algorithm to approximate the distributional flow from the learning dynamics in the mean field regime over weighted point clouds. In this setting, a contractive fixed point recursion computes the time-varying weights, numerically realizing the interacting Wasserstein gradient flow of the parameter distribution supported over the neuronal ensemble. An appealing aspect of the proposed algorithm is that the measure-valued recursions allow meshless computation. We demonstrate the proposed computational framework of interacting weighted particle evolution on binary and multi-class classification. Our algorithm performs gradient descent of the free energy associated with the risk functional.

We study recovery of amplitudes and nodes of a finite impulse train from a limited number of equispaced noisy frequency samples. This problem is known as super-resolution (SR) under sparsity constraints and has numerous applications, including direction of arrival and finite rate of innovation sampling. Prony's method is an algebraic technique which fully recovers the signal parameters in the absence of measurement noise. In the presence of noise, Prony's method may experience significant loss of accuracy, especially when the separation between Dirac pulses is smaller than the Nyquist-Shannon-Rayleigh (NSR) limit. In this work we combine Prony's method with a recently established decimation technique for analyzing the SR problem in the regime where the distance between two or more pulses is much smaller than the NSR limit. We show that our approach attains optimal asymptotic stability in the presence of noise. Our result challenges the conventional belief that Prony-type methods tend to be highly numerically unstable.

Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes by extending ideas of Chebop2 [Townsend and Olver, J. Comput. Phys., 299 (2015)] to the three-dimensional setting utilizing expansions in tensorized polynomial bases. Solving the discretized PDE involves a linear system that can be recast as a linear tensor equation. Under suitable additional assumptions, the structure of these equations admits for an efficient solution via the blocked recursive solver [Chen and Kressner, Numer. Algorithms, 84 (2020)]. In the general case, when these assumptions are not satisfied, this solver is used as a preconditioner to speed up computations.

We investigate the approximation formulas that were proposed by Tanaka & Sugihara (2019), in weighted Hardy spaces, which are analytic function spaces with certain asymptotic decay. Under the criterion of minimum worst error of $n$-point approximation formulas, we demonstrate that the formulas are nearly optimal. We also obtain the upper bounds of the approximation errors that coincide with the existing heuristic bounds in asymptotic order by duality theorem for the minimization problem of potential energy.

A burgeoning line of research has developed deep neural networks capable of approximating the solutions to high dimensional PDEs, opening related lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most theoretical analyses thus far have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as \emph{nonlinear elliptic variational PDEs}, whose solutions minimize an \emph{Euler-Lagrange} energy functional $\mathcal{E}(u) = \int_\Omega L(\nabla u) dx$. We show that if composing a function with Barron norm $b$ with $L$ produces a function of Barron norm at most $B_L b^p$, the solution to the PDE can be $\epsilon$-approximated in the $L^2$ sense by a function with Barron norm $O\left(\left(dB_L\right)^{p^{\log(1/\epsilon)}}\right)$. By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating $p, \epsilon, B_L$ as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs.

We study a discrete-time model where each packet has a cost of not being sent -- this cost might depend on the packet content. We study the tradeoff between the age and the cost where the sender is confined to packet-based strategies. The optimal tradeoff is found by an appropriate formulation of the problem as a Markov Decision Process (MDP). We show that the optimal tradeoff can be attained with finite-memory policies and we devise an efficient policy iteration algorithm to find these optimal policies. We further study a related problem where the transmitted packets are subject to erasures. We show that the optimal policies for our problem are also optimal for this new setup. Allowing coding across packets significantly extends the packet-based strategies. We show that when the packet payloads are small, the performance can be improved by coding.