Related papers: Frame theory in directional statistics
This manuscript studies statistical properties of linear classifiers obtained through minimization of an unregularized convex risk over a finite sample. Although the results are explicitly finite-dimensional, inputs may be passed through…
This survey explores the foundational theory and recent developments in the study of hyperuniformity. We present a comprehensive mathematical framework in the context of weakly stationary random measures, emphasizing spectral…
The hypothesis of randomness is fundamental in statistical machine learning and in many areas of nonparametric statistics; it says that the observations are assumed to be independent and coming from the same unknown probability…
We construct a new family of chance constrained directional models in stochastic data envelopment analysis, generalizing the deterministic directional models and the chance constrained radial models. We prove that chance constrained…
Binary density ratio estimation (DRE), the problem of estimating the ratio $p_1/p_2$ given their empirical samples, provides the foundation for many state-of-the-art machine learning algorithms such as contrastive representation learning…
The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These…
We investigate the problem of jointly testing a pair of composite hypotheses and, depending on the test result, estimating a random parameter under distributional uncertainties. Specifically, it is assumed that the distribution of the data…
Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment,…
Random feature maps are ubiquitous in modern statistical machine learning, where they generalize random projections by means of powerful, yet often difficult to analyze nonlinear operators. In this paper, we leverage the "concentration"…
In the uniformity testing task, an algorithm is provided with samples from an unknown probability distribution over a (known) finite domain, and must decide whether it is the uniform distribution, or, alternatively, if its total variation…
This paper investigates the consensus problem in almost sure sense for uncertain multi-agent systems with noises and fixed topology. By combining the tools of stochastic analysis, algebraic graph theory, and matrix theory, we analyze the…
Recently, diffusion probabilistic models (DPMs) have achieved promising results in diverse generative tasks. A typical DPM framework includes a forward process that gradually diffuses the data distribution and a reverse process that…
We develop large sample theory for merged data from multiple sources. Main statistical issues treated in this paper are (1) the same unit potentially appears in multiple datasets from overlapping data sources, (2) duplicated items are not…
Transition path theory computes statistics from ensembles of reactive trajectories. A common strategy for sampling reactive trajectories is to control the branching and pruning of trajectories so as to enhance the sampling of low…
In statistical problems, a set of parameterized probability distributions is used to estimate the true probability distribution. If Fisher information matrix at the true distribution is singular, then it has been left unknown what we can…
A theoretical framework is developed to describe the transformation that distributes probability density functions uniformly over space. In one dimension, the cumulative distribution can be used, but does not generalize to higher…
Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of…
Statistical pattern classification methods based on data-random graphs were introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the data points from various classes.…
In many real-world prediction tasks, class labels contain information about the relative order between labels that are not captured by commonly used loss functions such as multicategory cross-entropy. Recently, the preference for unimodal…
Ordinal measurements are common outcomes in studies within psychology, as well as in the social and behavioral sciences. Choosing an appropriate regression model for analysing such data poses a difficult task. This paper aims to facilitate…