Information and Inference-A Journal of the Ima最新文献

Matrix valued data has become increasingly prevalent in many applications. Most of the existing clustering methods for this type of data are tailored to the mean model and do not account for the dependence structure of the features, which can be very informative, especially in high-dimensional settings or when mean information is not available. To extract the information from the dependence structure for clustering, we propose a new latent variable model for the features arranged in matrix form, with some unknown membership matrices representing the clusters for the rows and columns. Under this model, we further propose a class of hierarchical clustering algorithms using the difference of a weighted covariance matrix as the dissimilarity measure. Theoretically, we show that under mild conditions, our algorithm attains clustering consistency in the high-dimensional setting. While this consistency result holds for our algorithm with a broad class of weighted covariance matrices, the conditions for this result depend on the choice of the weight. To investigate how the weight affects the theoretical performance of our algorithm, we establish the minimax lower bound for clustering under our latent variable model in terms of some cluster separation metric. Given these results, we identify the optimal weight in the sense that using this weight guarantees our algorithm to be minimax rate-optimal. The practical implementation of our algorithm with the optimal weight is also discussed. Simulation studies show that our algorithm performs better than existing methods in terms of the adjusted Rand index (ARI). The method is applied to a genomic dataset and yields meaningful interpretations.

Detecting and recovering a low-rank signal in a noisy data matrix is a fundamental task in data analysis. Typically, this task is addressed by inspecting and manipulating the spectrum of the observed data, e.g. thresholding the singular values of the data matrix at a certain critical level. This approach is well established in the case of homoskedastic noise, where the noise variance is identical across the entries. However, in numerous applications, the noise can be heteroskedastic, where the noise characteristics may vary considerably across the rows and columns of the data. In this scenario, the spectral behaviour of the noise can differ significantly from the homoskedastic case, posing various challenges for signal detection and recovery. To address these challenges, we develop an adaptive normalization procedure that equalizes the average noise variance across the rows and columns of a given data matrix. Our proposed procedure is data-driven and fully automatic, supporting a broad range of noise distributions, variance patterns and signal structures. Our approach relies on random matrix theory results that describe the resolvent of the noise via the so-called Dyson equation. By leveraging this relation, we can accurately infer the noise level in each row and each column directly from the resolvent of the data. We establish that in many cases, our normalization enforces the standard spectral behaviour of homoskedastic noise-the Marchenko-Pastur (MP) law, allowing for simple and reliable detection of signal components. Furthermore, we demonstrate that our approach can substantially improve signal recovery in heteroskedastic settings by manipulating the spectrum after normalization. Lastly, we apply our method to single-cell RNA sequencing and spatial transcriptomics data, showcasing accurate fits to the MP law after normalization.

Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn-Knopp (SK) iterations. This paper proves the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian with rates, when [Formula: see text] data points are i.i.d. sampled from a general [Formula: see text]-dimensional manifold embedded in a possibly high-dimensional space. Under certain joint limit of [Formula: see text] and kernel bandwidth [Formula: see text], the point-wise convergence rate of the graph Laplacian operator (under 2-norm) is proved to be [Formula: see text] at finite large [Formula: see text] up to log factors, achieved at the scaling of [Formula: see text]. When the manifold data are corrupted by outlier noise, we theoretically prove the graph Laplacian point-wise consistency which matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors among themselves and with data vectors. Motivated by our analysis, which suggests that not exact bi-stochastic normalization but an approximate one will achieve the same consistency rate, we propose an approximate and constrained matrix scaling problem that can be solved by SK iterations with early termination. Numerical experiments support our theoretical results and show the robustness of bi-stochastically normalized graph Laplacian to high-dimensional outlier noise.

We study the problem of estimating a [Formula: see text]-sparse signal [Formula: see text] from a set of noisy observations [Formula: see text] under the model [Formula: see text], where [Formula: see text] is the measurement matrix the row of which is drawn from distribution [Formula: see text]. We consider the class of [Formula: see text]-regularized least squares (LQLS) given by the formulation [Formula: see text], where [Formula: see text]  [Formula: see text] denotes the [Formula: see text]-norm. In the setting [Formula: see text] with fixed [Formula: see text] and [Formula: see text], we derive the asymptotic risk of [Formula: see text] for arbitrary covariance matrix [Formula: see text] that generalizes the existing results for standard Gaussian design, i.e. [Formula: see text]. The results were derived from the non-rigorous replica method. We perform a higher-order analysis for LQLS in the small-error regime in which the first dominant term can be used to determine the phase transition behavior of LQLS. Our results show that the first dominant term does not depend on the covariance structure of [Formula: see text] in the cases [Formula: see text] and [Formula: see text] which indicates that the correlations among predictors only affect the phase transition curve in the case [Formula: see text] a.k.a. LASSO. To study the influence of the covariance structure of [Formula: see text] on the performance of LQLS in the cases [Formula: see text] and [Formula: see text], we derive the explicit formulas for the second dominant term in the expansion of the asymptotic risk in terms of small error. Extensive computational experiments confirm that our analytical predictions are consistent with numerical results.

We consider asymptotically exact inference on the leading canonical correlation directions and strengths between two high-dimensional vectors under sparsity restrictions. In this regard, our main contribution is developing a novel representation of the Canonical Correlation Analysis problem, based on which one can operationalize a one-step bias correction on reasonable initial estimators. Our analytic results in this regard are adaptive over suitable structural restrictions of the high-dimensional nuisance parameters, which, in this set-up, correspond to the covariance matrices of the variables of interest. We further supplement the theoretical guarantees behind our procedures with extensive numerical studies.

Algorithmic stability is a concept from learning theory that expresses the degree to which changes to the input data (e.g. removal of a single data point) may affect the outputs of a regression algorithm. Knowing an algorithm's stability properties is often useful for many downstream applications-for example, stability is known to lead to desirable generalization properties and predictive inference guarantees. However, many modern algorithms currently used in practice are too complex for a theoretical analysis of their stability properties, and thus we can only attempt to establish these properties through an empirical exploration of the algorithm's behaviour on various datasets. In this work, we lay out a formal statistical framework for this kind of black-box testing without any assumptions on the algorithm or the data distribution, and establish fundamental bounds on the ability of any black-box test to identify algorithmic stability.