metric entropy estimation

Classical computational geometry algorithms often lack sensitivity to structural regularities in geometric inputs, leading to suboptimal performance on partially ordered or structured instances. Method: We introduce *range-partition entropy*, a unified input entropy measure that generalizes structural entropy from sorting to geometric problems—its first such extension. Leveraging this entropy, we design adaptive algorithms for fundamental tasks including 2D extreme points, 2D/3D convex hulls, and visibility queries. These algorithms integrate divide-and-conquer with preprocessing-based sorting to dynamically exploit local order in the input. Results: Our algorithms achieve entropy-sensitive running times—i.e., asymptotic complexity improves as input entropy decreases. Theoretical analysis shows they attain input-dependent optimal or near-optimal bounds for convex hulls and related problems, significantly outperforming traditional worst-case-optimal algorithms on structured inputs.

This work investigates the fundamental performance limits of learning and estimation tasks within an information-theoretic framework, independent of the computational capabilities of specific algorithms. By integrating tools from information theory and statistical learning theory—including metric entropy, VC dimension, Rademacher complexity, mutual information, and relative entropy—it systematically derives multiple upper bounds on generalization error. Simultaneously, leveraging Fano’s inequality together with covering and packing numbers, the study establishes information-theoretic lower bounds on minimax risk. The analysis unifies two complementary paradigms: one grounded in the geometric structure of metric spaces and the other based on information-theoretic measures. This synthesis yields a rigorous and broadly applicable theoretical framework for characterizing the optimal performance boundaries inherent to learning and estimation problems.

This work addresses the necessary and sufficient conditions for embedding a function space into an $L_p$-type reproducing kernel Banach space (RKBS). Method: By integrating functional analysis, metric entropy theory, and the geometric structure of RKBSs, the authors establish an exact characterization linking embeddability to the growth rate of metric entropy. Contribution/Results: They prove that a function space embeds into an $L_p$-type RKBS if and only if the metric entropy of its unit ball satisfies a specific upper bound—crucially, this bound alone is sufficient to guarantee existence of such an embedding. This reveals the universal modeling capacity of $L_p$-type RKBSs for function classes exhibiting controlled entropy growth. The result unifies and extends the theoretical scope of classical kernel methods, providing a novel analytical framework for learning high-dimensional, nonsmooth, and low-regularity functions.

This work investigates theoretical bounds on the size of metric balls under coordinate-wise additive metrics—particularly the sum-rank metric—to enhance the precision of error-correcting capability analysis in coding theory and network coding. Methodologically, it introduces the Boltzmann entropy method to ball-size analysis for such metrics, deriving universal entropy-based upper and lower bounds applicable to any coordinate-wise additive metric. For the sum-rank metric specifically, it establishes the first tight closed-form bounds, overcoming prior limitations that relied on numerical optimization or asymptotic approximations. Experimental evaluation demonstrates that, under typical parameter regimes, the new bounds improve bound tightness by 10%–35% over the state-of-the-art, thereby providing significantly stronger theoretical foundations for the design and performance evaluation of error-correcting codes.

This work addresses the sample complexity of Sample Average Approximation (SAA) for convex and strongly convex stochastic programming (SP) under standard SP assumptions—without requiring uniform Lipschitz continuity. Methodologically, it integrates convex analysis, stochastic optimization, and functional inequalities to bypass entropy-based arguments. The key contribution is the first tight, metric-entropy-free sample complexity bounds: $O(1/varepsilon^2)$ for convex SP and $O(1/varepsilon)$ for strongly convex SP—improving upon state-of-the-art bounds by a factor of $O(d)$ by eliminating dependence on covering numbers or Rademacher complexity. Theoretically, it reveals that SAA achieves nearly identical sample efficiency as stochastic mirror descent, bridging a long-standing gap in theoretical understanding. Numerical experiments validate the tightness of the bounds and demonstrate that SAA exhibits provably superior practical performance over stochastic mirror descent in non-Lipschitz settings.

This work investigates combinatorial bounds for codes in general finite metric spaces, with a focus on conditions under which the classical Gilbert–Varshamov (GV) bound can be surpassed. By modeling codes as independent sets in proximity graphs, the authors develop a generalized GV framework applicable to arbitrary metric spaces and introduce novel concepts such as Ramsey–Sidorenko graphs and independence-forcing graphs. Their analysis demonstrates that local subgraph statistics alone are insufficient to exceed the GV bound; instead, global structural properties of the space are essential. Leveraging tools from graph theory, extremal combinatorics, entropy optimization via KKT conditions, and fractional packing techniques, they derive code bounds for both vertex-transitive and non-edge-transitive graphs, establishing density thresholds for several graph families. In particular, they prove that in Hamming spaces, no improvement over the GV bound is possible using only local information, and provide a tight upper bound based on fractional packing.

This study addresses the problem of recovering true pairwise distances from noisy observations and identifying local clusters in general metric measure spaces under low regularity assumptions. To this end, the authors propose a near-linear-time algorithm that efficiently extracts large-scale local clusters around each sampled point at a fixed accuracy level for distance denoising, alongside a more computationally expensive algorithm achieving higher precision. This work is the first to extend distance denoising and clustering to general metric measure spaces, revealing a statistical–computational trade-off absent in the Riemannian manifold setting: while fixed-accuracy recovery can be achieved efficiently, attaining higher accuracy necessarily incurs greater computational cost.

This study addresses the parameterized Metrical Service Systems (MSS) problem, where request types are restricted to a known set of $m$ kinds. By modeling the problem via interval covering and employing a primal-dual approach, the authors design deterministic algorithms and establish matching adversarial lower bounds. On weighted star metrics, they achieve the first $O(m)$-competitive deterministic algorithm, matching the known randomized lower bound. On hierarchically separated trees (HSTs), they prove that no constant-competitive algorithm exists when $m \geq 4$, while presenting an $O(1)$-competitive deterministic algorithm for the case $m = 2$. This work resolves several open questions posed by Bubeck and Rabani, fully characterizing the performance limits of parameterized MSS on these two fundamental metric spaces.

This work re-examines whether neural networks genuinely overcome the curse of dimensionality, proposing computational bit complexity—rather than parameter count—as the central metric for assessing approximation efficiency. By integrating binary encoding and metric entropy theory, the authors develop a unified framework to systematically compare polynomial approximation, sparse grids, finite elements, and both shallow and deep neural networks. Their analysis reveals that the perceived advantages of neural networks stem not from an inherent superiority of their architecture, but from differences in the intrinsic complexity of the function classes they target. From the perspective of bit complexity, neural networks do not achieve approximation rates surpassing those of classical methods; the curse of dimensionality persists as a fundamental limitation rooted in bit complexity itself.