arXiv subjects

Yin-Ting Liao

Publications and source records attributed to Yin-Ting Liao.

3 recordsLinked to original sources

arXiv2023

Efficient and Robust Bayesian Selection of Hyperparameters in Dimension Reduction for Visualization

We introduce an efficient and robust auto-tuning framework for hyperparameter selection in dimension reduction (DR) algorithms, focusing on large-scale datasets and arbitrary performance metrics. By leveraging Bayesian optimization (BO) with a surrogate model, our approach enables efficient hyperparameter selection with multi-objective trade-offs and allows us to perform data-driven sensitivity analysis. By incorporating normalization and subsampling, the proposed framework demonstrates versatility and efficiency, as shown in applications to visualization techniques such as t-SNE and UMAP. We evaluate our results on various synthetic and real-world datasets using multiple quality metrics, providing a robust and efficient solution for hyperparameter selection in DR algorithms.

stat.ML↗

arXiv2023

Geometric sharp large deviations for random projections of $\ell_p^n$ spheres and balls

Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic exponential decay rate of probabilities, sharp large deviation estimates also provide the "prefactor" in front of the exponentially decaying term. For fixed $p \in (1,\infty)$, consider independent sequences $(X^{(n,p)})_{n \in \mathbb{N}}$ and $(Θ^n)_{n \in \mathbb{N}}$ of random vectors with $Θ^n$ distributed according to the normalized cone measure on the unit $\ell_2^n$ sphere, and $X^{(n,p)}$ distributed according to the normalized cone measure on the unit $\ell_p^n$ sphere. For almost every realization $(θ^n)_{n\in\mathbb{N}}$ of $(Θ^n)_{n\in\mathbb{N}}$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^{(n,p)}$ onto $θ^n$, that are asymptotically exact (as the dimension $n$ tends to infinity). Furthermore, the case when $(X^{(n,p)})_{n \in \mathbb{N}}$ is replaced with $(\mathscr{X}^{(n,p)})_{n \in \mathbb{N}}$, where $\mathscr{X}^{(n,p)}$ is distributed according to the uniform (or normalized volume) measure on the unit $\ell_p^n$ ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions $(θ^n)_{n \in\mathbb{N}}$ that encodes additional geometric information that enables one to distinguish between projections of balls and spheres. Moreover, comparison with numerical estimates obtained by direct computation and importance sampling shows that the obtained analytical expressions for tail probabilities provide good approximations even for moderate values of $n$.

math.PR↗

arXiv2021

An asymptotic thin shell condition and large deviations for random multidimensional projections

Consider the projection of an $n$-dimensional random vector onto a random $k_n$-dimensional basis, $k_n \leq n$, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal $k_n$-frames in $\mathbb{R}^n$, in three different asymptotic regimes as $n \rightarrow \infty$: "constant" ($k_n=k$), "sublinear" ($k_n \rightarrow \infty$ but $k_n/n \rightarrow 0$) and "linear" $k_n/n \rightarrow λ$ with $0 < λ\le 1$). When the sequence of random vectors satisfies a certain "asymptotic thin shell condition", we establish annealed large deviation principles (LDPs) for the corresponding sequence of random projections in the constant regime, and for the sequence of empirical measures of the coordinates of the random projections in the sublinear and linear regimes. We also establish LDPs for certain scaled $\ell_q$ norms of the random projections in these different regimes. Moreover, we verify our assumptions for various sequences of random vectors of interest, including those distributed according to Gibbs measures with superquadratic interaction potential, or the uniform measure on suitably scaled $\ell_p^n$ balls, for $p \in [1,\infty)$, and generalized Orlicz balls defined via a superquadratic function. Our results complement the central limit theorem for convex sets and related results which are known to hold under a "thin shell" condition. These results also substantially extend existing large deviation results for random projections, which are first, restricted to the setting of measures on $\ell_p^n$ balls, and secondly, limited to univariate LDPs (i.e., in $\mathbb{R}$) involving either the norm of a $k_n$-dimensional projection or the projection of $X^{(n)}$ onto a random one-dimensional subspace. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.

math.PR↗