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We consider supervised learning (regression/classification) problems with tensor-valued input. We derive multi-linear sufficient reductions for the regression or classification problem by modeling the conditional distribution of the predictors given the response as a member of the quadratic exponential family. We develop estimation procedures of sufficient reductions for both continuous and binary tensor-valued predictors. We prove the consistency and asymptotic normality of the estimated sufficient reduction using manifold theory. For continuous predictors, the estimation algorithm is highly computationally efficient and is also applicable to situations where the dimension of the reduction exceeds the sample size. We demonstrate the superior performance of our approach in simulations and real-world data examples for both continuous and binary tensor-valued predictors.

We derive exact upper and lower bounds for the cumulative distribution function (cdf) of the output of a neural network over its entire support subject to noisy (stochastic) inputs. The upper and lower bounds converge to the true cdf over its domain as the resolution increases. Our method applies to any feedforward NN using continuous monotonic piecewise differentiable activation functions (e.g., ReLU, tanh and softmax) and convolutional NNs, which were beyond the scope of competing approaches. The novelty and an instrumental tool of our approach is to bound general NNs with ReLU NNs. The ReLU NN based bounds are then used to derive upper and lower bounds of the cdf of the NN output. Experiments demonstrate that our method delivers guaranteed bounds of the predictive output distribution over its support, thus providing exact error guarantees, in contrast to competing approaches., Comment: 16 pages

The property of shadowing has been shown to be fundamental in both the theory of symbolic dynamics as well as continuous dynamical systems. A quintessential class of discontinuous dynamical systems are those driven by transitive piecewise monotone interval maps and in particular $\beta$-transformations, namely transformations of the form $T_{\beta, \alpha} : x \mapsto \beta x + \alpha \; (\operatorname{mod} \, 1)$ acting on $[0,1]$. We provide a short elegant proof showing that this class of dynamical systems does not possess the property of shadowing, complementing and extending the work of Chen and Portela., Comment: 5 pages

Generative AI is transforming education by enabling personalized learning, enhancing administrative efficiency, and fostering creative engagement. This paper explores the opportunities and challenges these tools bring to pedagogy, proposing actionable frameworks to address existing equity gaps. Ethical considerations such as algorithmic bias, data privacy, and AI role in human centric education are emphasized. The findings underscore the need for responsible AI integration that ensures accessibility, equity, and innovation in educational systems., Comment: 12 pages

The synergy of strong spin-orbit coupling and electron-electron interactions gives rise to unconventional topological states, such as topological Mott insulator, Weyl semimetal, and quantum spin liquid. In this study, we have grown single crystals of the pyrochlore iridate Pr$_2$Ir$_2$O$_7$ and explored its magnetic, lattice dynamical, and electronic properties. While Raman spectroscopy data reveal six phonon modes confirming the cubic \textit{Fd$\bar{3}$m} crystal symmetry, dc magnetic susceptibility data show no anomalies and hence indicate the absence of magnetic phase transitions down to 2~K. Both temperature-dependent electric transport and optical conductivity data reveal the metallic character of Pr$_2$Ir$_2$O$_7$. The optical conductivity spectrum contains a mid-infrared absorption band, which becomes more pronounced with decreasing temperature due to spectral weight transfer from high to low energies. The presence of the mid-infrared band hints at the importance of correlation physics. The optical response furthermore suggests that Pr$_2$Ir$_2$O$_7$ is close to the Weyl semimetal phase., Comment: 11 pages, 8 figures, accepted for publication in Phys. Rev. B

We address the resource allocation challenges in NextG cellular radio access networks (RAN), where heterogeneous user applications demand guarantees on throughput and service regularity. We leverage the Whittle indexability property to decompose the resource allocation problem, enabling the independent computation of relative priorities for each user. By simply allocating resources in decreasing order of these indices, we transform the combinatorial complexity of resource allocation into a linear one. We propose Windex, a lightweight approach for training neural networks to compute Whittle indices, considering constraint violation, channel quality, and system load. Implemented on a real-time RAN intelligent controller (RIC), our approach enables resource allocation decision times of less than 20$\mu$s per user and efficiently allocates resources in each 1ms scheduling time slot. Evaluation across standardized 3GPP service classes demonstrates significant improvements in service guarantees compared to existing schedulers, validated through simulations and emulations with over-the-air channel traces on a 5G testbed., Comment: 15 pages, 8 figures

Media streaming is the dominant application over wireless edge (access) networks. The increasing softwarization of such networks has led to efforts at intelligent control, wherein application-specific actions may be dynamically taken to enhance the user experience. The goal of this work is to develop and demonstrate learning-based policies for optimal decision making to determine which clients to dynamically prioritize in a video streaming setting. We formulate the policy design question as a constrained Markov decision problem (CMDP), and observe that by using a Lagrangian relaxation we can decompose it into single-client problems. Further, the optimal policy takes a threshold form in the video buffer length, which enables us to design an efficient constrained reinforcement learning (CRL) algorithm to learn it. Specifically, we show that a natural policy gradient (NPG) based algorithm that is derived using the structure of our problem converges to the globally optimal policy. We then develop a simulation environment for training, and a real-world intelligent controller attached to a WiFi access point for evaluation. We empirically show that the structured learning approach enables fast learning. Furthermore, such a structured policy can be easily deployed due to low computational complexity, leading to policy execution taking only about 15$\mu$s. Using YouTube streaming experiments in a resource constrained scenario, we demonstrate that the CRL approach can increase quality of experience (QOE) by over 30\%., Comment: 15 pages, 14 figures

A new perspective is introduced regarding the analysis of Multiple Sequence Alignments (MSA), representing aligned data defined over a finite alphabet of symbols. The framework is designed to produce a block decomposition of an MSA, where each block is comprised of sequences exhibiting a certain site-coherence. The key component of this framework is an information theoretical potential defined on pairs of sites (links) within the MSA. This potential quantifies the expected drop in variation of information between the two constituent sites, where the expectation is taken with respect to all possible sub-alignments, obtained by removing a finite, fixed collection of rows. It is proved that the potential is zero for linked sites representing columns, whose symbols are in bijective correspondence and it is strictly positive, otherwise. It is furthermore shown that the potential assumes its unique minimum for links at which each symbol pair appears with the same multiplicity. Finally, an application is presented regarding anomaly detection in an MSA, composed of inverse fold solutions of a fixed tRNA secondary structure, where the anomalies are represented by inverse fold solutions of a different RNA structure., Comment: 15 pages, 2 figures

Many stochastic continuous-state dynamical systems can be modeled as probabilistic programs with nonlinear non-polynomial updates in non-nested loops. We present two methods, one approximate and one exact, to automatically compute, without sampling, moment-based invariants for such probabilistic programs as closed-form solutions parameterized by the loop iteration. The exact method applies to probabilistic programs with trigonometric and exponential updates and is embedded in the Polar tool. The approximate method for moment computation applies to any nonlinear random function as it exploits the theory of polynomial chaos expansion to approximate non-polynomial updates as the sum of orthogonal polynomials. This translates the dynamical system to a non-nested loop with polynomial updates, and thus renders it conformable with the Polar tool that computes the moments of any order of the state variables. We evaluate our methods on an extensive number of examples ranging from modeling monetary policy to several physical motion systems in uncertain environments. The experimental results demonstrate the advantages of our approach with respect to the current state-of-the-art., Comment: Published in ACM Transactions on Modeling and Computer Simulation (TOMACS). Extended version of the conference paper 'Moment-based Invariants for Probabilistic Loops with Non-polynomial Assignments' published at QEST 2022 (Best paper award, see also the preprint arxiv.org/abs/2205.02577). arXiv admin note: substantial text overlap with arXiv:2205.02577

Probabilistic loops can be employed to implement and to model different processes ranging from software to cyber-physical systems. One main challenge is how to automatically estimate the distribution of the underlying continuous random variables symbolically and without sampling. We develop an approach, which we call K-series estimation, to approximate statically the joint and marginal distributions of a vector of random variables updated in a probabilistic non-nested loop with polynomial and non-polynomial assignments. Our approach is a general estimation method for an unknown probability density function with bounded support. It naturally complements algorithms for automatic derivation of moments in probabilistic loops such as~\cite{BartocciKS19,Moosbruggeretal2022}. Its only requirement is a finite number of moments of the unknown density. We show that Gram-Charlier (GC) series, a widely used estimation method, is a special case of K-series when the normal probability density function is used as reference distribution. We provide also a formulation suitable for estimating both univariate and multivariate distributions. We demonstrate the feasibility of our approach using multiple examples from the literature., Comment: 34 pages

We propose a novel mathematical paradigm for the study of genetic variation in sequence alignments. This framework originates from extending the notion of pairwise relations, upon which current analysis is based on, to k-ary dissimilarity. This dissimilarity naturally leads to a generalization of simplicial complexes by endowing simplices with weights, compatible with the boundary operator. We introduce the notion of k-stances and dissimilarity complex, the former encapsulating arithmetic as well as topological structure expressing these k-ary relations. We study basic mathematical properties of dissimilarity complexes and show how this approach captures an entirely new layer of biologically relevant viral dynamics in the context of SARS-CoV-2 and H1N1 flu genomic data.

We provide a bottom up construction of torsion generators for weighted homology of a weighted complex over a discrete valuation ring $R=\mathbb{F}[[\pi]]$. This is achieved by starting from a basis for classical homology of the $n$-th skeleton for the underlying complex with coefficients in the residue field $\mathbb{F}$ and then lifting it to a basis for the weighted homology with coefficients in the ring $R$. Using the latter, a bijection is established between $n+1$ and $n$ dimensional simplices whose weight ratios provide the exponents of the $\pi$-monomials that generate each torsion summand in the structure theorem of the weighted homology modules over $R$. We present algorithms that subsume the torsion computation by reducing it to normalization over the residue field of $R$, and describe a Python package we implemented that takes advantage of this reduction and performs the computation efficiently., Comment: 16 pages, 2 figures

We present an algorithmic approach to estimate the value distributions of random variables of probabilistic loops whose statistical moments are (partially) known. Based on these moments, we apply two statistical methods, Maximum Entropy and Gram-Charlier series, to estimate the distributions of the loop's random variables. We measure the accuracy of our distribution estimation by comparing the resulting distributions using exact and estimated moments of the probabilistic loop, and performing statistical tests. We evaluate our method on several probabilistic loops with polynomial updates over random variables drawing from common probability distributions, including examples implementing financial and biological models. For this, we leverage symbolic approaches to compute exact higher-order moments of loops as well as use sampling-based techniques to estimate moments from loop executions. Our experimental results provide practical evidence of the accuracy of our method for estimating distributions of probabilistic loop outputs.

We present a method to automatically approximate moment-based invariants of probabilistic programs with non-polynomial updates of continuous state variables to accommodate more complex dynamics. Our approach leverages polynomial chaos expansion to approximate non-linear functional updates as sums of orthogonal polynomials. We exploit this result to automatically estimate state-variable moments of all orders in Prob-solvable loops with non-polynomial updates. We showcase the accuracy of our estimation approach in several examples, such as the turning vehicle model and the Taylor rule in monetary policy., Comment: 23 pages