This paper simplifies computing higher-order U-statistics efficiently.
problem The inefficiency of computing higher-order U-statistics in practice. method Decomposition, connection to Einstein summation, and treewidth-based complexity estimate.
result A new, more efficient algorithm to compute U-statistics. We study the fundamental tradeoffs between computational tractability and statistical accuracy for a general family of hypothesis testing problems with combinatorial structures. Based upon an oracle model of computation, which captures the interactions between algorithms and data, we establish a general lower bound tha…
Establishes statistical and computational bounds for influence diagnostics.
problem Identifying influential datapoints or subsets in machine learning models.
method Finite-sample statistical bounds and computational complexity for influence functions and approximate maximum influence perturbations.
result Established statistical and computational guarantees for influence diagnostics.
Noise Sensitivity Exponent controls statistical-computational gaps in learning.
problem Understanding when learning is statistically possible yet computationally hard in high-dimensional statistics.
method Investigating statistical-computational gaps in single- and multi-index models using Noise Sensitivity Exponent.
result Noise Sensitivity Exponent governs statistical-computational gaps in high-dimensional learning.
Survey on using low-degree polynomials to assess statistical tasks complexity.
problem Understanding the complexity of statistical tasks using polynomial functions.
method Applying low-degree polynomials to measure the complexity of statistical tasks, including detection, recovery, and estimation.
result Low-degree polynomials provide a framework to predict and explain statistical-computational tradeoffs.
Improved statistical computation through efficient matrix sampling.
problem Reducing computational cost in large-scale statistical methods.
method Accumulative sub-sampling method to improve statistical efficiency.
result Effective matrix size control improves computational efficiency.
Nyström KPCA balances computational efficiency and statistical accuracy.
problem Computational burden in large sample situations for kernel methods.
method Theoretical analysis of Nyström approximate kernel principal component analysis (KPCA).
result Nyström approximate KPCA matches statistical performance of non-approximate KPCA while being computationally beneficial.
Study trade-offs between statistical and computational efficiency in variational inference.
problem Optimizing statistical accuracy vs. computational efficiency in Bayesian inference.
method Case study on Gaussian inferential models with diagonal plus low-rank precision matrices, analyzing Bayesian posterior inference and frequentist uncertainty quantification errors.
result Lower-rank models reduce variance and accelerate convergence but increase posterior inference error.
Approximate Bayesian Computation (ABC) methods are used to approximate posterior distributions in models with unknown or computationally intractable likelihoods. Both the accuracy and computational efficiency of ABC depend on the choice of summary statistic, but outside of special cases where the optimal summary statis…
Improved computational complexity in statistical models using second-order information.
problem Polynomial convergence of gradient descent in singular statistical models.
method Normalized Gradient Descent (NormGD) algorithm with second-order information.
result NormGD reaches final statistical radius in logarithmic iterations of n. New computational lower bounds for clustering and related problems.
problem Statistical-computational gaps in high-dimensional clustering problems.
method Investigation of low-degree polynomials in latent space models to derive lower bounds.
result New and sharper computational lower bounds for clustering, sparse clustering, and biclustering.
New method combines score lists using joint CDFs, improving computation.
problem Combining non-comparable score lists over a common index set.
method New algorithm for computing joint CDF values, linear runtime.
result Improved computation of joint CDF values for N-dimensional order statistics.
Study shows computational and statistical gaps in Gaussian Single-Index Models.
problem Statistical and computational trade-offs in high-dimensional regression problems.
method Analysis of SQ and LDP frameworks, partial-trace algorithm.
result Computational algorithms require significantly more samples than information-theoretic limits.
A research frontier has emerged in scientific computation, wherein numerical error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly dete…
In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems. These are regimes in which the underlying statistical problem is information-theoretically possible although no efficient algorithm exists, rendering the problem essentially unsolvable for large instances…
A new method uses neural tangent kernel to efficiently compute MMD statistic.
problem Efficiently computing Maximum Mean Discrepancy (MMD) statistic with low memory and computational complexity.
method Identifies a connection between neural tangent kernel (NTK) and MMD to develop a computationally and memory-efficient approach.
result The proposed NTK-MMD statistic is validated through numerical experiments on synthetic and real-world datasets.
New methods improve statistical accuracy of complex models without high computational cost.
problem Improving statistical accuracy of complex models without high computational cost.
method Neural posterior and likelihood estimation (NPE and NLE) methods.
result NPE and NLE methods have similar theoretical guarantees to ABC and BSL, but achieve accuracy at a reduced computational cost.
Efficient method for tensor linear form inference with noisy incomplete data.
problem Statistical inference of tensor linear forms with incomplete and noisy observations.
method Initial estimate + debiasing + one-step power iteration.
result Optimal uncertainty quantification and statistical-to-computational gaps examined.
Statistical query algorithms and low-degree tests are nearly equivalent in high-dimensional hypothesis testing.
problem High-dimensional hypothesis testing and information-computation gaps.
method Analysis of statistical query framework and low-degree polynomials.
result Statistical query algorithms and low-degree polynomials are almost equivalent in power under mild conditions.
Paper explores limits of high-order clustering with planted structures.
problem Statistical and computational limits of high-order clustering with planted structures.
method Developed methods for detection and recovery of clusters, identified signal-to-noise ratio boundaries.
result Sharp boundaries of signal-to-noise ratio for statistical and computational feasibility.
Approximate Bayesian computation is an established and popular method for likelihood-free inference with applications in many disciplines. The effectiveness of the method depends critically on the availability of well performing summary statistics. Summary statistic selection relies heavily on domain knowledge and care…
Optimizes ICA performance in high dimensions with computational constraints.
problem Statistical optimality and computational tractability in ICA.
method Characterization of optimal sample complexity, development of computationally tractable estimates.
result Optimal sample complexity is linear in dimensionality, quadratic with low-degree polynomial algorithms.
We consider the weakly supervised binary classification problem where the labels are randomly flipped with probability 1−α. Although there exist numerous algorithms for this problem, it remains theoretically unexplored how the statistical accuracies and computational efficiency of these algorithms depend on the degr…
Proposes a new method to improve Bayesian computation accuracy using flexible classification.
problem Bayesian computations accuracy check using rank-based simulation-based calibration has limitations.
method Replaces marginal rank test with a flexible classification approach that learns from data.
result Improves statistical power and provides an interpretable divergence measure of miscalibration.
Big Data bring new opportunities to modern society and challenges to data scientists. On one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data intro…
This article reviews statistical methods for learning data representations.
problem Learning meaningful representations of data.
method Statistical perspective on unsupervised and supervised representation learning.
result Recent advances in representation learning from a statistical viewpoint.
We investigate the efficiency of k-means in terms of both statistical and computational requirements. More precisely, we study a Nyström approach to kernel k-means. We analyze the statistical properties of the proposed method and show that it achieves the same accuracy of exact kernel k-means with only a fraction of co…
Sliced Optimal Transport simplifies OT for fast computation.
problem Efficient computation of distances and barycenters for probability measures.
method Combines OT, integral geometry, and statistics for fast computation.
result Retains rich geometric structure while speeding up computations.
New insights link diverse statistical problems via secret leakage planted clique.
problem Statistical-computational gaps in inference problems.
method Secret leakage planted clique as a new hardness assumption for reductions.
result Establishes tight statistical-computational tradeoffs for various problems.
Paper studies statistical-computational trade-offs in tensor PCA and related problems.
problem Statistical-computational gap in tensor PCA estimation.
method Derives computational lower bounds using communication complexity.
result Lower bounds specify trade-off among passes, sample size, and memory.
Modern technologies are generating ever-increasing amounts of data. Making use of these data requires methods that are both statistically sound and computationally efficient. Typically, the statistical and computational aspects are treated separately. In this paper, we propose an approach to entangle these two aspects …
How should statistical procedures be designed so as to be scalable computationally to the massive datasets that are increasingly the norm? When coupled with the requirement that an answer to an inferential question be delivered within a certain time budget, this question has significant repercussions for the field of s…
This article is the rejoinder for the paper "Probabilistic Integration: A Role in Statistical Computation?" to appear in Statistical Science with discussion. We would first like to thank the reviewers and many of our colleagues who helped shape this paper, the editor for selecting our paper for discussion, and of cours…
Extends JKO scheme for iterative algorithms with unknown parameters.
problem Computational and statistical analysis of iterative algorithms with unknown parameters.
method Develops statistical methods to estimate unknown parameters and adapts JKO scheme.
result Establishes asymptotic theory for the statistical JKO scheme.
Convolutional neural networks learn effective summary statistics for ABC inference.
problem Selecting high-quality summary statistics for accurate ABC inference in complex systems.
method Proposes a CNN architecture to automatically learn informative summary statistics from time series data.
result CNNs can effectively circumvent the statistics selection problem in ABC inference.
Active learning method for ABC statistics selection reduces expert work and improves posterior estimates.
problem Handling intractable likelihood functions in models with domain knowledge.
method Active learning method for selecting summary statistics in ABC.
result Better posterior estimates than existing methods, especially with limited simulation budget.
Optimized Franz-Parisi criterion matches SQ lower bounds for various statistical models.
problem Understanding computational hardness in statistical inference.
method Proposed and refined Franz-Parisi criterion, established equivalence with SQ lower bounds.
result Optimized Franz-Parisi criterion is equivalent to Statistical Query (SQ) lower bounds.
We discuss the relative merits of optimistic and randomized approaches to exploration in reinforcement learning. Optimistic approaches presented in the literature apply an optimistic boost to the value estimate at each state-action pair and select actions that are greedy with respect to the resulting optimistic value f…
Statistical-computational gap found in aligning multiple Gaussian graphs.
problem Aligning multiple Gaussian graphs with unknown signals.
method Generalized informational threshold and computational barrier analysis.
result Existence of a statistical-computational gap in multiple Gaussian graph alignment.
Paper introduces a method to assess the statistical reliability of changepoints using selective inference and dynamic programming.
problem Assessing the statistical reliability of detected changepoints.
method Selective inference framework combined with dynamic programming for exact p-value computation.
result Proposes a method with high statistical power and decent computational efficiency.
This paper investigates asymptotic behaviors of gradient descent algorithms (particularly accelerated gradient descent and stochastic gradient descent) in the context of stochastic optimization arising in statistics and machine learning where objective functions are estimated from available data. We show that these alg…
The paper tackles statistical and computational challenges in learning correlated reward models.
problem The Independence of Irrelevant Alternatives (IIA) assumption collapses human preferences into a universal utility function, leading to coarse approximations.
method The paper investigates the statistical and computational challenges of learning a correlated probit model using best-of-three preference data.
result Best-of-three preference data overcomes the limitations of pairwise preference data, allowing for more fine-grained modeling of human preferences.
The scalability of statistical estimators is of increasing importance in modern applications. One approach to implementing scalable algorithms is to compress data into a low dimensional latent space using dimension reduction methods. In this paper we develop an approach for dimension reduction that exploits the assumpt…
Database theory and database practice are typically the domain of computer scientists who adopt what may be termed an algorithmic perspective on their data. This perspective is very different than the more statistical perspective adopted by statisticians, scientific computers, machine learners, and other who work on wh…
The development of cluster computing frameworks has allowed practitioners to scale out various statistical estimation and machine learning algorithms with minimal programming effort. This is especially true for machine learning problems whose objective function is nicely separable across individual data points, such as…
The interplay between computational efficiency and statistical accuracy in high-dimensional inference has drawn increasing attention in the literature. In this paper, we study computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple non-overlapping) signal submatrix…
U-statistics improve gradient estimation in importance-weighted variational inference.
problem High variance in gradient estimation for importance-weighted variational inference.
method Use U-statistics to average base gradient estimators on overlapping batches of size m, achieving lower variance.
result U-statistic variance reduction leads to modest to significant improvements in inference performance.
Polynomial-time algorithm finds planted hypercube vectors in Gaussian mixtures.
problem Clustering d-dimensional Gaussian mixtures with unknown covariance.
method Lattice-based methods using Lenstra--Lenstra--Lovasz reduction.
result Achieves statistically-optimal sample complexity of d+1 samples.