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We introduce KANICE (Kolmogorov-Arnold Networks with Interactive Convolutional Elements), a novel neural architecture that combines Convolutional Neural Networks (CNNs) with Kolmogorov-Arnold Network (KAN) principles. KANICE integrates Interactive Convolutional Blocks (ICBs) and KAN linear layers into a CNN framework. This leverages KANs' universal approximation capabilities and ICBs' adaptive feature learning. KANICE captures complex, non-linear data relationships while enabling dynamic, context-dependent feature extraction based on the Kolmogorov-Arnold representation theorem. We evaluated KANICE on four datasets: MNIST, Fashion-MNIST, EMNIST, and SVHN, comparing it against standard CNNs, CNN-KAN hybrids, and ICB variants. KANICE consistently outperformed baseline models, achieving 99.35% accuracy on MNIST and 90.05% on the SVHN dataset. Furthermore, we introduce KANICE-mini, a compact variant designed for efficiency. A comprehensive ablation study demonstrates that KANICE-mini achieves comparable performance to KANICE with significantly fewer parameters. KANICE-mini reached 90.00% accuracy on SVHN with 2,337,828 parameters, compared to KANICE's 25,432,000. This study highlights the potential of KAN-based architectures in balancing performance and computational efficiency in image classification tasks. Our work contributes to research in adaptive neural networks, integrates mathematical theorems into deep learning architectures, and explores the trade-offs between model complexity and performance, advancing computer vision and pattern recognition. The source code for this paper is publicly accessible through our GitHub repository (https://github.com/m-ferdaus/kanice).

Mixed multinomial logits are discrete mixtures introduced several decades ago to model the probability of choosing an attribute from $p$ possible candidates, in heterogeneous populations. The model has recently attracted attention in the AI literature, under the name softmax mixtures, where it is routinely used in the final layer of a neural network to map a large number $p$ of vectors in $\mathbb{R}^L$ to a probability vector. Despite its wide applicability and empirical success, statistically optimal estimators of the mixture parameters, obtained via algorithms whose running time scales polynomially in $L$, are not known. This paper provides a solution to this problem for contemporary applications, such as large language models, in which the mixture has a large number $p$ of support points, and the size $N$ of the sample observed from the mixture is also large. Our proposed estimator combines two classical estimators, obtained respectively via a method of moments (MoM) and the expectation-minimization (EM) algorithm. Although both estimator types have been studied, from a theoretical perspective, for Gaussian mixtures, no similar results exist for softmax mixtures for either procedure. We develop a new MoM parameter estimator based on latent moment estimation that is tailored to our model, and provide the first theoretical analysis for a MoM-based procedure in softmax mixtures. Although consistent, MoM for softmax mixtures can exhibit poor numerical performance, as observed other mixture models. Nevertheless, as MoM is provably in a neighborhood of the target, it can be used as warm start for any iterative algorithm. We study in detail the EM algorithm, and provide its first theoretical analysis for softmax mixtures. Our final proposal for parameter estimation is the EM algorithm with a MoM warm start.

We propose a procedure for estimating the Schr\"odinger bridge between two probability distributions. Unlike existing approaches, our method does not require iteratively simulating forward and backward diffusions or training neural networks to fit unknown drifts. Instead, we show that the potentials obtained from solving the static entropic optimal transport problem between the source and target samples can be modified to yield a natural plug-in estimator of the time-dependent drift that defines the bridge between two measures. Under minimal assumptions, we show that our proposal, which we call the \emph{Sinkhorn bridge}, provably estimates the Schr\"odinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure. Our approach combines results from the areas of sampling, and theoretical and statistical entropic optimal transport., Comment: 39 pages, 3 figures, 1 table

The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension $d$ of the problem increases, the number of iterations required to ensure convergence within a desired error in the $W_2$ metric scales in proportion to $d$ or $\sqrt{d}$. In this paper, we argue that, despite this poor scaling of the $W_2$ error for the full set of variables, the behavior for a small number of variables can be significantly better: a number of iterations proportional to $K$, up to logarithmic terms in $d$, often suffices for the algorithm to converge to within a desired $W_2$ error for all $K$-marginals. We refer to this effect as delocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. Our analysis relies on a novel $W_{2,\ell^\infty}$ metric to measure convergence. A key technical challenge we address is the lack of a one-step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.

Imbalanced datasets are a significant challenge in real-world scenarios. They lead to models that underperform on underrepresented classes, which is a critical issue in infrastructure inspection. This paper introduces the Enhanced Feature Pyramid Network (E-FPN), a deep learning model for the semantic segmentation of culverts and sewer pipes within imbalanced datasets. The E-FPN incorporates architectural innovations like sparsely connected blocks and depth-wise separable convolutions to improve feature extraction and handle object variations. To address dataset imbalance, the model employs strategies like class decomposition and data augmentation. Experimental results on the culvert-sewer defects dataset and a benchmark aerial semantic segmentation drone dataset show that the E-FPN outperforms state-of-the-art methods, achieving an average Intersection over Union (IoU) improvement of 13.8% and 27.2%, respectively. Additionally, class decomposition and data augmentation together boost the model's performance by approximately 6.9% IoU. The proposed E-FPN presents a promising solution for enhancing object segmentation in challenging, multi-class real-world datasets, with potential applications extending beyond culvert-sewer defect detection.

We prove bounds on statistical distances between high-dimensional exchangeable mixture distributions (which we call permutation mixtures) and their i.i.d. counterparts. Our results are based on a novel method for controlling $\chi^2$ divergences between exchangeable mixtures, which is tighter than the existing methods of moments or cumulants. At a technical level, a key innovation in our proofs is a new Maclaurin-type inequality for elementary symmetric polynomials of variables that sum to zero and an upper bound on permanents of doubly-stochastic positive semidefinite matrices. Our results imply a de Finetti-style theorem (in the language of Diaconis and Freedman, 1987) and general asymptotic results for compound decision problems, generalizing and strengthening a result of Hannan and Robbins (1955)., Comment: Added the reference Tang and Polyanskiy (2023) and fixed some typos

This paper introduces Semantic Haar-Adaptive Refined Pyramid Network (SHARP-Net), a novel architecture for semantic segmentation. SHARP-Net integrates a bottom-up pathway featuring Inception-like blocks with varying filter sizes (3x3$ and 5x5), parallel max-pooling, and additional spatial detection layers. This design captures multi-scale features and fine structural details. Throughout the network, depth-wise separable convolutions are used to reduce complexity. The top-down pathway of SHARP-Net focuses on generating high-resolution features through upsampling and information fusion using $1\times1$ and $3\times3$ depth-wise separable convolutions. We evaluated our model using our developed challenging Culvert-Sewer Defects dataset and the benchmark DeepGlobe Land Cover dataset. Our experimental evaluation demonstrated the base model's (excluding Haar-like features) effectiveness in handling irregular defect shapes, occlusions, and class imbalances. It outperformed state-of-the-art methods, including U-Net, CBAM U-Net, ASCU-Net, FPN, and SegFormer, achieving average improvements of 14.4% and 12.1% on the Culvert-Sewer Defects and DeepGlobe Land Cover datasets, respectively, with IoU scores of 77.2% and 70.6%. Additionally, the training time was reduced. Furthermore, the integration of carefully selected and fine-tuned Haar-like features enhanced the performance of deep learning models by at least 20%. The proposed SHARP-Net, incorporating Haar-like features, achieved an impressive IoU of 94.75%, representing a 22.74% improvement over the base model. These features were also applied to other deep learning models, showing a 35.0% improvement, proving their versatility and effectiveness. SHARP-Net thus provides a powerful and efficient solution for accurate semantic segmentation in challenging real-world scenarios.

We present an introduction to the field of statistical optimal transport, based on lectures given at \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XLIX., Comment: Lecture Notes for \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XLIX 2019

Optimal transport (OT) has profoundly impacted machine learning by providing theoretical and computational tools to realign datasets. In this context, given two large point clouds of sizes $n$ and $m$ in $\mathbb{R}^d$, entropic OT (EOT) solvers have emerged as the most reliable tool to either solve the Kantorovich problem and output a $n\times m$ coupling matrix, or to solve the Monge problem and learn a vector-valued push-forward map. While the robustness of EOT couplings/maps makes them a go-to choice in practical applications, EOT solvers remain difficult to tune because of a small but influential set of hyperparameters, notably the omnipresent entropic regularization strength $\varepsilon$. Setting $\varepsilon$ can be difficult, as it simultaneously impacts various performance metrics, such as compute speed, statistical performance, generalization, and bias. In this work, we propose a new class of EOT solvers (ProgOT), that can estimate both plans and transport maps. We take advantage of several opportunities to optimize the computation of EOT solutions by dividing mass displacement using a time discretization, borrowing inspiration from dynamic OT formulations, and conquering each of these steps using EOT with properly scheduled parameters. We provide experimental evidence demonstrating that ProgOT is a faster and more robust alternative to standard solvers when computing couplings at large scales, even outperforming neural network-based approaches. We also prove statistical consistency of our approach for estimating optimal transport maps., Comment: In proceedings of the 38th conference on Advances in Neural Information Processing Systems (NeurIPs), 23 pages, 14 figures

The rapid progress in Large Language Models (LLMs) could transform many fields, but their fast development creates significant challenges for oversight, ethical creation, and building user trust. This comprehensive review looks at key trust issues in LLMs, such as unintended harms, lack of transparency, vulnerability to attacks, alignment with human values, and environmental impact. Many obstacles can undermine user trust, including societal biases, opaque decision-making, potential for misuse, and the challenges of rapidly evolving technology. Addressing these trust gaps is critical as LLMs become more common in sensitive areas like finance, healthcare, education, and policy. To tackle these issues, we suggest combining ethical oversight, industry accountability, regulation, and public involvement. AI development norms should be reshaped, incentives aligned, and ethics integrated throughout the machine learning process, which requires close collaboration across technology, ethics, law, policy, and other fields. Our review contributes a robust framework to assess trust in LLMs and analyzes the complex trust dynamics in depth. We provide contextualized guidelines and standards for responsibly developing and deploying these powerful AI systems. This review identifies key limitations and challenges in creating trustworthy AI. By addressing these issues, we aim to build a transparent, accountable AI ecosystem that benefits society while minimizing risks. Our findings provide valuable guidance for researchers, policymakers, and industry leaders striving to establish trust in LLMs and ensure they are used responsibly across various applications for the good of society., Comment: Under review at Proceedings of the IEEE

Entropic Brenier maps are regularized analogues of Brenier maps (optimal transport maps) which converge to Brenier maps as the regularization parameter shrinks. In this work, we prove quantitative stability bounds between entropic Brenier maps under variations of the target measure. In particular, when all measures have bounded support, we establish the optimal Lipschitz constant for the mapping from probability measures to entropic Brenier maps. This provides an exponential improvement to a result of Carlier, Chizat, and Laborde (2024). As an application, we prove near-optimal bounds for the stability of semi-discrete \emph{unregularized} Brenier maps for a family of discrete target measures.

One of the products of the National Solar Observatory's Integrated Synoptic Program (NISP) is the farside seismic map which shows the magnetic activity on the unobserved side of the Sun. The production of these rudimentary maps began in 2006, and they have since proven to be a valuable tool in tracking solar activity which cannot be directly observed from the earth's surface. The continuous tracking of solar active regions allows space weather forecasters to monitor critical solar events which may have larger economic and societal impacts here on Earth. In an effort to improve these maps, several steps are underway through the Windows on the Universe project (WoU) funded by the NSF. One of these steps is to improve the quality assurance measures for the images collected at individual sites throughout the GONG network and is used to develop the farside maps. To this end, we have designed a binary classification neural network to determine which of these site images should and should not be included in the farside pipeline that produces the end product maps. This convolutional neural network is a highly effective and computationally efficient method of significantly improving the quality of the farside maps currently produced by the NISP program., Comment: 31 Pages (including 27 figures and 6 tables)

The proposed architecture, Dual Attentive U-Net with Feature Infusion (DAU-FI Net), addresses challenges in semantic segmentation, particularly on multiclass imbalanced datasets with limited samples. DAU-FI Net integrates multiscale spatial-channel attention mechanisms and feature injection to enhance precision in object localization. The core employs a multiscale depth-separable convolution block, capturing localized patterns across scales. This block is complemented by a spatial-channel squeeze and excitation (scSE) attention unit, modeling inter-dependencies between channels and spatial regions in feature maps. Additionally, additive attention gates refine segmentation by connecting encoder-decoder pathways. To augment the model, engineered features using Gabor filters for textural analysis, Sobel and Canny filters for edge detection are injected guided by semantic masks to expand the feature space strategically. Comprehensive experiments on a challenging sewer pipe and culvert defect dataset and a benchmark dataset validate DAU-FI Net's capabilities. Ablation studies highlight incremental benefits from attention blocks and feature injection. DAU-FI Net achieves state-of-the-art mean Intersection over Union (IoU) of 95.6% and 98.8% on the defect test set and benchmark respectively, surpassing prior methods by 8.9% and 12.6%, respectively. Ablation studies highlight incremental benefits from attention blocks and feature injection. The proposed architecture provides a robust solution, advancing semantic segmentation for multiclass problems with limited training data. Our sewer-culvert defects dataset, featuring pixel-level annotations, opens avenues for further research in this crucial domain. Overall, this work delivers key innovations in architecture, attention, and feature engineering to elevate semantic segmentation efficacy., Comment: under review in IEEE Transactions on Artificial Intelligence

One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law onto any other given law. A line of recent work has analyzed $L^2$ convergence rates of plugin estimators of Brenier maps, which are defined as the Brenier map between density estimators of the underlying distributions. In this work, we show that such estimators satisfy a pointwise central limit theorem when the underlying laws are supported on the flat torus of dimension $d \geq 3$. We also derive a negative result, showing that these estimators do not converge weakly in $L^2$ when the dimension is sufficiently large. Our proofs hinge upon a quantitative linearization of the Monge-Amp\`ere equation, which may be of independent interest. This result allows us to reduce our problem to that of deriving limit laws for the solution of a uniformly elliptic partial differential equation with a stochastic right-hand side, subject to periodic boundary conditions.

This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples., Comment: 79 pages

Given a source and a target probability measure supported on $\mathbb{R}^d$, the Monge problem asks to find the most efficient way to map one distribution to the other. This efficiency is quantified by defining a \textit{cost} function between source and target data. Such a cost is often set by default in the machine learning literature to the squared-Euclidean distance, $\ell^2_2(\mathbf{x},\mathbf{y})=\tfrac12|\mathbf{x}-\mathbf{y}|_2^2$. Recently, Cuturi et. al '23 highlighted the benefits of using elastic costs, defined through a regularizer $\tau$ as $c(\mathbf{x},\mathbf{y})=\ell^2_2(\mathbf{x},\mathbf{y})+\tau(\mathbf{x}-\mathbf{y})$. Such costs shape the \textit{displacements} of Monge maps $T$, i.e., the difference between a source point and its image $T(\mathbf{x})-\mathbf{x})$, by giving them a structure that matches that of the proximal operator of $\tau$. In this work, we make two important contributions to the study of elastic costs: (i) For any elastic cost, we propose a numerical method to compute Monge maps that are provably optimal. This provides a much-needed routine to create synthetic problems where the ground truth OT map is known, by analogy to the Brenier theorem, which states that the gradient of any convex potential is always a valid Monge map for the $\ell_2^2$ cost; (ii) We propose a loss to \textit{learn} the parameter $\theta$ of a parameterized regularizer $\tau_\theta$, and apply it in the case where $\tau_{A}(\mathbf{z})=|A^\perp \mathbf{z}|^2_2$. This regularizer promotes displacements that lie on a low dimensional subspace of $\mathbb{R}^d$, spanned by the $p$ rows of $A\in\mathbb{R}^{p\times d}$.

Current WHO guidelines set prevalence thresholds below which a Neglected Tropical Disease can be considered to have been eliminated as a public health problem, and specify how surveys to assess whether elimination has been achieved should be designed and analysed, based on classical survey sampling methods. In this paper we describe an alternative approach based on geospatial statistical modelling. We first show the gains in efficiency that can be obtained by exploiting any spatial correlation in the underlying prevalence surface. We then suggest that the current guidelines implicit use of a significance testing argument is not appropriate; instead, we argue for a predictive inferential framework, leading to design criteria based on controlling the rates at which areas whose true prevalence lies above and below the elimination threshold are incorrectly classified. We describe how this approach naturally accommodates context-specific information in the form of georeferenced covariates that have been shown to be predictive of disease prevalence. Finally, we give a progress report of an ongoing collaboration with the Guyana Ministry of Health Neglected Tropical Disease program on the design of an IDA (Ivermectin, Diethylcarbamazine and Albendazole) Impact Survey (IIS) of lymphatic filariasis to be conducted in Guyana in early 2023, Comment: 11 pages, 3 figures

We study the consistency of surrogate risks for robust binary classification. It is common to learn robust classifiers by adversarial training, which seeks to minimize the expected $0$-$1$ loss when each example can be maliciously corrupted within a small ball. We give a simple and complete characterization of the set of surrogate loss functions that are \emph{consistent}, i.e., that can replace the $0$-$1$ loss without affecting the minimizing sequences of the original adversarial risk, for any data distribution. We also prove a quantitative version of adversarial consistency for the $\rho$-margin loss. Our results reveal that the class of adversarially consistent surrogates is substantially smaller than in the standard setting, where many common surrogates are known to be consistent., Comment: 17 pages, published in NeurIps 2023. version 3: added acknowledgements, no other changes. version 2: reorganized Section 4 and added proofs of the approximate complimentary slackness theorems. arXiv admin note: text overlap with arXiv:2206.09099

A major question in the study of the Erd\H{o}s--R\'enyi random graph is to understand the probability that it contains a given subgraph. This study originated in classical work of Erd\H{o}s and R\'enyi (1960). More recent work studies this question both in building a general theory of sharp versus coarse transitions (Friedgut and Bourgain 1999; Hatami, 2012) and in results on the location of the transition (Kahn and Kalai, 2007; Talagrand, 2010; Frankston, Kahn, Narayanan, Park, 2019; Park and Pham, 2022). In inference problems, one often studies the optimal accuracy of inference as a function of the amount of noise. In a variety of sparse recovery problems, an ``all-or-nothing (AoN) phenomenon'' has been observed: Informally, as the amount of noise is gradually increased, at some critical threshold the inference problem undergoes a sharp jump from near-perfect recovery to near-zero accuracy (Gamarnik and Zadik, 2017; Reeves, Xu, Zadik, 2021). We can regard AoN as the natural inference analogue of the sharp threshold phenomenon in random graphs. In contrast with the general theory developed for sharp thresholds of random graph properties, the AoN phenomenon has only been studied so far in specific inference settings. In this paper we study the general problem of inferring a graph $H=H_n$ planted in an Erd\H{o}s--R\'enyi random graph, thus naturally connecting the two lines of research mentioned above. We show that questions of AoN are closely connected to first moment thresholds, and to a generalization of the so-called Kahn--Kalai expectation threshold that scans over subgraphs of $H$ of edge density at least $q$. In a variety of settings we characterize AoN, by showing that AoN occurs if and only if this ``generalized expectation threshold'' is roughly constant in $q$. Our proofs combine techniques from random graph theory and Bayesian inference., Comment: 41 pages