The sole aim of this book is to give a self-contained introduction to concepts and mathematical tools in Bayesian matrix decomposition in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning Bayesian matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of variational inference for conducting the optimization. We refer the reader to literature in the field of Bayesian analysis for a more detailed introduction to the related fields. This book is primarily a summary of purpose, significance of important Bayesian matrix decomposition methods, e.g., real-valued decomposition, nonnegative matrix factorization, Bayesian interpolative decomposition, and the origin and complexity of the methods which shed light on their applications. The mathematical prerequisite is a first course in statistics and linear algebra. Other than this modest background, the development is self-contained, with rigorous proof provided throughout.
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A series of recent works by Lyu, Wang, Vadhan, and Zhang (TCC `21, NeurIPS `22, STOC `23) showed that composition theorems for non-interactive differentially private mechanisms extend to the concurrent composition of interactive differentially private mechanism, when differential privacy is measured using $f$-DP and the adversary is adaptive. We extend their work to the $\textit{continual observation setting,}$ where the data is arriving online in a potentially adaptive manner. More specifically, we show that all composition theorems for non-interactive differentially private mechanisms extend to the concurrent composition of continual differentially private mechanism, where the adversary is adaptive. We show this result for $f$-DP, which also implies the result for pure DP and $(\epsilon, \delta)$-DP.
We consider the problem of estimating the spectrum of a symmetric bounded entry (not necessarily PSD) matrix via entrywise sampling. This problem was introduced by [Bhattacharjee, Dexter, Drineas, Musco, Ray '22], where it was shown that one can obtain an $\epsilon n$ additive approximation to all eigenvalues of $A$ by sampling a principal submatrix of dimension $\frac{\text{poly}(\log n)}{\epsilon^3}$. We improve their analysis by showing that it suffices to sample a principal submatrix of dimension $\tilde{O}(\frac{1}{\epsilon^2})$ (with no dependence on $n$). This matches known lower bounds and therefore resolves the sample complexity of this problem up to $\log\frac{1}{\epsilon}$ factors. Using similar techniques, we give a tight $\tilde{O}(\frac{1}{\epsilon^2})$ bound for obtaining an additive $\epsilon\|A\|_F$ approximation to the spectrum of $A$ via squared row-norm sampling, improving on the previous best $\tilde{O}(\frac{1}{\epsilon^{8}})$ bound. We also address the problem of approximating the top eigenvector for a bounded entry, PSD matrix $A.$ In particular, we show that sampling $O(\frac{1}{\epsilon})$ columns of $A$ suffices to produce a unit vector $u$ with $u^T A u \geq \lambda_1(A) - \epsilon n$. This matches what one could achieve via the sampling bound of [Musco, Musco'17] for the special case of approximating the top eigenvector, but does not require adaptivity. As additional applications, we observe that our sampling results can be used to design a faster eigenvalue estimation sketch for dense matrices resolving a question of [Swartworth, Woodruff'23], and can also be combined with [Musco, Musco'17] to achieve $O(1/\epsilon^3)$ (adaptive) sample complexity for approximating the spectrum of a bounded entry PSD matrix to $\epsilon n$ additive error.
Validity, reliability, and fairness are core ethical principles embedded in classical argument-based assessment validation theory. These principles are also central to the Standards for Educational and Psychological Testing (2014) which recommended best practices for early applications of artificial intelligence (AI) in high-stakes assessments for automated scoring of written and spoken responses. Responsible AI (RAI) principles and practices set forth by the AI ethics community are critical to ensure the ethical use of AI across various industry domains. Advances in generative AI have led to new policies as well as guidance about the implementation of RAI principles for assessments using AI. Building on Chapelle's foundational validity argument work to address the application of assessment validation theory for technology-based assessment, we propose a unified assessment framework that considers classical test validation theory and assessment-specific and domain-agnostic RAI principles and practice. The framework addresses responsible AI use for assessment that supports validity arguments, alignment with AI ethics to maintain human values and oversight, and broader social responsibility associated with AI use.
Gaussian graphical regressions have emerged as a powerful approach for regressing the precision matrix of a Gaussian graphical model on covariates, which, unlike traditional Gaussian graphical models, can help determine how graphs are modulated by high dimensional subject-level covariates, and recover both the population-level and subject-level graphs. To fit the model, a multi-task learning approach {achieves} %has been shown to result in lower error rates compared to node-wise regressions. However, due to the high complexity and dimensionality of the Gaussian graphical regression problem, the important task of statistical inference remains unexplored. We propose a class of debiased estimators based on multi-task learners for statistical inference in Gaussian graphical regressions. We show that debiasing can be performed quickly and separately for the multi-task learners. In a key debiasing step {that estimates} %involving the estimation of the inverse covariance matrix, we propose a novel {projection technique} %diagonalization approach that dramatically reduces computational costs {in optimization} to scale only with the sample size $n$. We show that our debiased estimators enjoy a fast convergence rate and asymptotically follow a normal distribution, enabling valid statistical inference such as constructing confidence intervals and performing hypothesis testing. Simulation studies confirm the practical utility of the proposed approach, and we further apply it to analyze gene co-expression graph data from a brain cancer study, revealing meaningful biological relationships.
Kernel methods underpin many of the most successful approaches in data science and statistics, and they allow representing probability measures as elements of a reproducing kernel Hilbert space without loss of information. Recently, the kernel Stein discrepancy (KSD), which combines Stein's method with the flexibility of kernel techniques, gained considerable attention. Through the Stein operator, KSD allows the construction of powerful goodness-of-fit tests where it is sufficient to know the target distribution up to a multiplicative constant. However, the typical U- and V-statistic-based KSD estimators suffer from a quadratic runtime complexity, which hinders their application in large-scale settings. In this work, we propose a Nystr\"om-based KSD acceleration -- with runtime $\mathcal O\left(mn+m^3\right)$ for $n$ samples and $m\ll n$ Nystr\"om points -- , show its $\sqrt{n}$-consistency with a classical sub-Gaussian assumption, and demonstrate its applicability for goodness-of-fit testing on a suite of benchmarks.
We introduce Statistical Flow Matching (SFM), a novel and mathematically rigorous flow-matching framework on the manifold of parameterized probability measures inspired by the results from information geometry. We demonstrate the effectiveness of our method on the discrete generation problem by instantiating SFM on the manifold of categorical distributions whose geometric properties remain unexplored in previous discrete generative models. Utilizing the Fisher information metric, we equip the manifold with a Riemannian structure whose intrinsic geometries are effectively leveraged by following the shortest paths of geodesics. We develop an efficient training and sampling algorithm that overcomes numerical stability issues with a diffeomorphism between manifolds. Our distinctive geometric perspective of statistical manifolds allows us to apply optimal transport during training and interpret SFM as following the steepest direction of the natural gradient. Unlike previous models that rely on variational bounds for likelihood estimation, SFM enjoys the exact likelihood calculation for arbitrary probability measures. We manifest that SFM can learn more complex patterns on the statistical manifold where existing models often fail due to strong prior assumptions. Comprehensive experiments on real-world generative tasks ranging from image, text to biological domains further demonstrate that SFM achieves higher sampling quality and likelihood than other discrete diffusion or flow-based models.
Solving algebra problems (APs) continues to attract significant research interest as evidenced by the large number of algorithms and theories proposed over the past decade. Despite these important research contributions, however, the body of work remains incomplete in terms of theoretical justification and scope. The current contribution intends to fill the gap by developing a review framework that aims to lay a theoretical base, create an evaluation scheme, and extend the scope of the investigation. This paper first develops the State Transform Theory (STT), which emphasizes that the problem-solving algorithms are structured according to states and transforms unlike the understanding that underlies traditional surveys which merely emphasize the progress of transforms. The STT, thus, lays the theoretical basis for a new framework for reviewing algorithms. This new construct accommodates the relation-centric algorithms for solving both word and diagrammatic algebra problems. The latter not only highlights the necessity of introducing new states but also allows revelation of contributions of individual algorithms obscured in prior reviews without this approach.
Humans perceive the world by concurrently processing and fusing high-dimensional inputs from multiple modalities such as vision and audio. Machine perception models, in stark contrast, are typically modality-specific and optimised for unimodal benchmarks, and hence late-stage fusion of final representations or predictions from each modality (`late-fusion') is still a dominant paradigm for multimodal video classification. Instead, we introduce a novel transformer based architecture that uses `fusion bottlenecks' for modality fusion at multiple layers. Compared to traditional pairwise self-attention, our model forces information between different modalities to pass through a small number of bottleneck latents, requiring the model to collate and condense the most relevant information in each modality and only share what is necessary. We find that such a strategy improves fusion performance, at the same time reducing computational cost. We conduct thorough ablation studies, and achieve state-of-the-art results on multiple audio-visual classification benchmarks including Audioset, Epic-Kitchens and VGGSound. All code and models will be released.
Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.
Embedding entities and relations into a continuous multi-dimensional vector space have become the dominant method for knowledge graph embedding in representation learning. However, most existing models ignore to represent hierarchical knowledge, such as the similarities and dissimilarities of entities in one domain. We proposed to learn a Domain Representations over existing knowledge graph embedding models, such that entities that have similar attributes are organized into the same domain. Such hierarchical knowledge of domains can give further evidence in link prediction. Experimental results show that domain embeddings give a significant improvement over the most recent state-of-art baseline knowledge graph embedding models.
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