TBA

TBA

I will discuss a circle of questions about the geometry of sums of large sets in R^n endowed with the Gaussian measure, related to the convexity problem of Talagrand: if a subset A of R^n has Gaussian measure at least 2/3 say, for which integer q does A+...+A (q times) contain a large convex body? I will explain why it is closely connected to the problem of understanding sums of non-independent Gaussian random vectors. I will discuss some sharp results and partial answers.

Multidimensional scaling (MDS) has a long history in statistics and underpins a broad class of unsupervised learning and spectral/nonlinear dimension reduction techniques. The objective of MDS is to extract meaningful information from relational data (e.g., distances between sensors, correlations between brain regions, or disagreement scores between individuals) by embedding these relationships into a Euclidean space. In practice, the observed relational information is often subject to measurement errors and/or corrupted by noise. However, the resulting embeddings are typically interpreted as exploratory visualizations without accounting for these variations. This talk presents recent work developing a principled statistical framework for MDS. We show that the classical MDS algorithm achieves minimax-optimal performance across a wide range of noise models and loss functions. Building on this, we develop a framework for constructing valid confidence sets for the embedded points obtained via MDS, enabling formal uncertainty quantification for geometric structure inferred from noisy relational data. These results provide a theoretical foundation for interpreting MDS embeddings, and extend naturally to a wide range of unsupervised learning techniques in modern data science.

In modern machine learning applications, data matrices are always assumed to admit the signal-plus-noise structure. Typically, we assume that the spectra of signal and noise matrices are well-separated and that noise subspaces only produce a marginal influence. While these assumptions are readily verified for dense matrices via classical random matrix theory, real-world data is often sparse, posing significant theoretical challenges. We investigate the spectral properties of a matrix $X \in \mathbb{R}^{n \times m}$ with i.i.d. $\text{Bernoulli}(p)$ entries. Previous literature proved that when $np \gg \log(n)$, the singular values of $X$ almost surely remain within the compact support of the Marčenko-Pastur (MP) distribution. However, we identify a critical sparsity regime $p = b \log(n) / \sqrt{mn}$ where this classical result fails. We provide a quantitative characterization of the emergence of outlier singular values. For explicit thresholds $b_*$ and $b^*$ as functions of the aspect ratio $\gamma = n/m \ge 1$, we prove a three-phase transition: (1) for $b > b_*$, no outliers exist; (2) for $b^* < b < b_*$, outliers emerge only beyond the right edge of the MP law; and (3) for $b < b^*$, outliers appear on both sides of the bulk, all with high probability. The locations of those outliers are precisely determined by the largest and smallest degree vertices of the underlying random graph. Besides, behavior of singular vectors corresponding to bulk and edge singular values can be characterized precisely. Our approach follows the framework established by Alt, Ducatez, and Knowles (2021), which can be extended to sparse matrices with general bounded entries.