Tusn\'ady's problem asks to bound the discrepancy of points and axis-parallel boxes in $\mathbb{R}^d$. Algorithmic bounds on Tusn\'ady's problem use a canonical decomposition of Matou\v{s}ek for the system of points and axis-parallel boxes, together with other techniques like partial coloring and / or random-walk based methods. We use the notion of \emph{shallow cell complexity} and the \emph{shallow packing lemma}, together with the chaining technique, to obtain an improved decomposition of the set system. Coupled with an algorithmic technique of Bansal and Garg for discrepancy minimization, which we also slightly extend, this yields improved algorithmic bounds on Tusn\'ady's problem. For $d\geq 5$, our bound matches the lower bound of $\Omega(\log^{d-1}n)$ given by Matou\v{s}ek, Nikolov and Talwar [IMRN, 2020] -- settling Tusn\'ady's problem, upto constant factors. For $d=2,3,4$, we obtain improved algorithmic bounds of $O(\log^{7/4}n)$, $O(\log^{5/2}n)$ and $O(\log^{13/4}n)$ respectively, which match or improve upon the non-constructive bounds of Nikolov for $d\geq 3$. Further, we also give improved bounds for the discrepancy of set systems of points and polytopes in $\mathbb{R}^d$ generated via translations of a fixed set of hyperplanes. As an application, we also get a bound for the geometric discrepancy of anchored boxes in $\mathbb{R}^d$ with respect to an arbitrary measure, matching the upper bound for the Lebesgue measure, which improves on a result of Aistleitner, Bilyk, and Nikolov [MC and QMC methods, \emph{Springer, Proc. Math. Stat.}, 2018] for $d\geq 4$.
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The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces. The pieces must be placed so that in the resulting placement, they are pairwise interior-disjoint. We establish a framework which enables us to show that for many combinations of allowed pieces, containers and motions, the resulting problem is $\exists \mathbb{R}$-complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that it is an $\exists \mathbb{R}$-complete problem to decide if a set of convex polygons, each of which has at most $7$ corners, can be packed into a square. Restricted to translations, we show that the following problems are $\exists \mathbb{R}$-complete: (i) pieces bounded by segments and hyperbolic curves to be packed in a square, and (ii) convex polygons to be packed in a container bounded by segments and hyperbolic curves.
The hard thresholding technique plays a vital role in the development of algorithms for sparse signal recovery. By merging this technique and heavy-ball acceleration method which is a multi-step extension of the traditional gradient descent method, we propose the so-called heavy-ball-based hard thresholding (HBHT) and heavy-ball-based hard thresholding pursuit (HBHTP) algorithms for signal recovery. It turns out that the HBHT and HBHTP can successfully recover a $k$-sparse signal if the restricted isometry constant of the measurement matrix satisfies $\delta_{3k}<0.618 $ and $\delta_{3k}<0.577,$ respectively. The guaranteed success of HBHT and HBHTP is also shown under the conditions $\delta_{2k}<0.356$ and $\delta_{2k}<0.377,$ respectively. Moreover, the finite convergence and stability of the two algorithms are also established in this paper. Simulations on random problem instances are performed to compare the performance of the proposed algorithms and several existing ones. Empirical results indicate that the HBHTP performs very comparably to a few existing algorithms and it takes less average time to achieve the signal recovery than these existing methods.
We consider the problem of training a classification model with group annotated training data. Recent work has established that, if there is distribution shift across different groups, models trained using the standard empirical risk minimization (ERM) objective suffer from poor performance on minority groups and that group distributionally robust optimization (Group-DRO) objective is a better alternative. The starting point of this paper is the observation that though Group-DRO performs better than ERM on minority groups for some benchmark datasets, there are several other datasets where it performs much worse than ERM. Inspired by ideas from the closely related problem of domain generalization, this paper proposes a new and simple algorithm that explicitly encourages learning of features that are shared across various groups. The key insight behind our proposed algorithm is that while Group-DRO focuses on groups with worst regularized loss, focusing instead, on groups that enable better performance even on other groups, could lead to learning of shared/common features, thereby enhancing minority performance beyond what is achieved by Group-DRO. Empirically, we show that our proposed algorithm matches or achieves better performance compared to strong contemporary baselines including ERM and Group-DRO on standard benchmarks on both minority groups and across all groups. Theoretically, we show that the proposed algorithm is a descent method and finds first order stationary points of smooth nonconvex functions.
We consider the question of adaptive data analysis within the framework of convex optimization. We ask how many samples are needed in order to compute $\epsilon$-accurate estimates of $O(1/\epsilon^2)$ gradients queried by gradient descent, and we provide two intermediate answers to this question. First, we show that for a general analyst (not necessarily gradient descent) $\Omega(1/\epsilon^3)$ samples are required. This rules out the possibility of a foolproof mechanism. Our construction builds upon a new lower bound (that may be of interest of its own right) for an analyst that may ask several non adaptive questions in a batch of fixed and known $T$ rounds of adaptivity and requires a fraction of true discoveries. We show that for such an analyst $\Omega (\sqrt{T}/\epsilon^2)$ samples are necessary. Second, we show that, under certain assumptions on the oracle, in an interaction with gradient descent $\tilde \Omega(1/\epsilon^{2.5})$ samples are necessary. Our assumptions are that the oracle has only \emph{first order access} and is \emph{post-hoc generalizing}. First order access means that it can only compute the gradients of the sampled function at points queried by the algorithm. Our assumption of \emph{post-hoc generalization} follows from existing lower bounds for statistical queries. More generally then, we provide a generic reduction from the standard setting of statistical queries to the problem of estimating gradients queried by gradient descent. These results are in contrast with classical bounds that show that with $O(1/\epsilon^2)$ samples one can optimize the population risk to accuracy of $O(\epsilon)$ but, as it turns out, with spurious gradients.
Given a set $P$ of $n$ points in the plane, the $k$-center problem is to find $k$ congruent disks of minimum possible radius such that their union covers all the points in $P$. The $2$-center problem is a special case of the $k$-center problem that has been extensively studied in the recent past \cite{CAHN,HT,SH}. In this paper, we consider a generalized version of the $2$-center problem called \textit{proximity connected} $2$-center (PCTC) problem. In this problem, we are also given a parameter $\delta\geq 0$ and we have the additional constraint that the distance between the centers of the disks should be at most $\delta$. Note that when $\delta=0$, the PCTC problem is reduced to the $1$-center(minimum enclosing disk) problem and when $\delta$ tends to infinity, it is reduced to the $2$-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 \cite{ACN0}, but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic $O(n^2\log n)$ time algorithm for the problem.
We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the \emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $\mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.
We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems, which we call dynamical dimension reduction (DDR). In the DDR model, each point is evolved via a nonlinear flow towards a lower-dimensional subspace; the projection onto the subspace gives the low-dimensional embedding. Training the model involves identifying the nonlinear flow and the subspace. Following the equation discovery method, we represent the vector field that defines the flow using a linear combination of dictionary elements, where each element is a pre-specified linear/nonlinear candidate function. A regularization term for the average total kinetic energy is also introduced and motivated by optimal transport theory. We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method. We also show how the DDR method can be trained using a gradient-based optimization method, where the gradients are computed using the adjoint method from optimal control theory. The DDR method is implemented and compared on synthetic and example datasets to other dimension reductions methods, including PCA, t-SNE, and Umap.
Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.
Similarity query is the family of queries based on some similarity metrics. Unlike the traditional database queries which are mostly based on value equality, similarity queries aim to find targets "similar enough to" the given data objects, depending on some similarity metric, e.g., Euclidean distance, cosine similarity and so on. To measure the similarity between data objects, traditional methods normally work on low level or syntax features(e.g., basic visual features on images or bag-of-word features of text), which makes them weak to compute the semantic similarities between objects. So for measuring data similarities semantically, neural embedding is applied. Embedding techniques work by representing the raw data objects as vectors (so called "embeddings" or "neural embeddings" since they are mostly generated by neural network models) that expose the hidden semantics of the raw data, based on which embeddings do show outstanding effectiveness on capturing data similarities, making it one of the most widely used and studied techniques in the state-of-the-art similarity query processing research. But there are still many open challenges on the efficiency of embedding based similarity query processing, which are not so well-studied as the effectiveness. In this survey, we first provide an overview of the "similarity query" and "similarity query processing" problems. Then we talk about recent approaches on designing the indexes and operators for highly efficient similarity query processing on top of embeddings (or more generally, high dimensional data). Finally, we investigate the specific solutions with and without using embeddings in selected application domains of similarity queries, including entity resolution and information retrieval. By comparing the solutions, we show how neural embeddings benefit those applications.
Edge intelligence refers to a set of connected systems and devices for data collection, caching, processing, and analysis in locations close to where data is captured based on artificial intelligence. The aim of edge intelligence is to enhance the quality and speed of data processing and protect the privacy and security of the data. Although recently emerged, spanning the period from 2011 to now, this field of research has shown explosive growth over the past five years. In this paper, we present a thorough and comprehensive survey on the literature surrounding edge intelligence. We first identify four fundamental components of edge intelligence, namely edge caching, edge training, edge inference, and edge offloading, based on theoretical and practical results pertaining to proposed and deployed systems. We then aim for a systematic classification of the state of the solutions by examining research results and observations for each of the four components and present a taxonomy that includes practical problems, adopted techniques, and application goals. For each category, we elaborate, compare and analyse the literature from the perspectives of adopted techniques, objectives, performance, advantages and drawbacks, etc. This survey article provides a comprehensive introduction to edge intelligence and its application areas. In addition, we summarise the development of the emerging research field and the current state-of-the-art and discuss the important open issues and possible theoretical and technical solutions.
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