arXiv subjects

Lee-Ad Gottlieb

Publications and source records attributed to Lee-Ad Gottlieb.

At least 19 recordsLinked to original sources

arXiv2025

Improved fixed-parameter bounds for Min-Sum-Radii and Diameters $k$-clustering and their fair variants

We provide improved upper and lower bounds for the Min-Sum-Radii (MSR) and Min-Sum-Diameters (MSD) clustering problems with a bounded number of clusters $k$. In particular, we propose an exact MSD algorithm with running-time $n^{O(k)}$. We also provide $(1+ε)$ approximation algorithms for both MSR and MSD with running-times of $O(kn) +(1/ε)^{O(dk)}$ in metrics spaces of doubling dimension $d$. Our algorithms extend to $k$-center, improving upon previous results, and to $α$-MSR, where radii are raised to the $α$ power for $α>1$. For $α$-MSD we prove an exponential time ETH-based lower bound for $α>\log 3$. All algorithms can also be modified to handle outliers. Moreover, we can extend the results to variants that observe fairness constraints, as well as to the general framework of mergeable clustering, which includes many other popular clustering variants. We complement these upper bounds with ETH-based lower bounds for these problems, in particular proving that $n^{O(k)}$ time is tight for MSR and $α$-MSR even in doubling spaces, and that $2^{o(k)}$ bounds are impossible for MSD.

cs.DS↗

arXiv2023

Weighted Distance Nearest Neighbor Condensing

The problem of nearest neighbor condensing has enjoyed a long history of study, both in its theoretical and practical aspects. In this paper, we introduce the problem of weighted distance nearest neighbor condensing, where one assigns weights to each point of the condensed set, and then new points are labeled based on their weighted distance nearest neighbor in the condensed set. We study the theoretical properties of this new model, and show that it can produce dramatically better condensing than the standard nearest neighbor rule, yet is characterized by generalization bounds almost identical to the latter. We then suggest a condensing heuristic for our new problem. We demonstrate Bayes consistency for this heuristic, and also show promising empirical results.

cs.LG↗

arXiv2023

Labelings vs. Embeddings: On Distributed Representations of Distances

We investigate for which metric spaces the performance of distance labeling and of $\ell_\infty$-embeddings differ, and how significant can this difference be. Recall that a distance labeling is a distributed representation of distances in a metric space $(X,d)$, where each point $x\in X$ is assigned a succinct label, such that the distance between any two points $x,y \in X$ can be approximated given only their labels. A highly structured special case is an embedding into $\ell_\infty$, where each point $x\in X$ is assigned a vector $f(x)$ such that $\|f(x)-f(y)\|_\infty$ is approximately $d(x,y)$. The performance of a distance labeling or an $\ell_\infty$-embedding is measured via its distortion and its label-size/dimension. We also study the analogous question for the prioritized versions of these two measures. Here, a priority order $π=(x_1,\dots,x_n)$ of the point set $X$ is given, and higher-priority points should have shorter labels. Formally, a distance labeling has prioritized label-size $α(\cdot)$ if every $x_j$ has label size at most $α(j)$. Similarly, an embedding $f: X \to \ell_\infty$ has prioritized dimension $α(\cdot)$ if $f(x_j)$ is non-zero only in the first $α(j)$ coordinates. In addition, we compare these prioritized measures to their classical (worst-case) versions. We answer these questions in several scenarios, uncovering a surprisingly diverse range of behaviors. First, in some cases labelings and embeddings have very similar worst-case performance, but in other cases there is a huge disparity. However in the prioritized setting, we most often find a strict separation between the performance of labelings and embeddings. And finally, when comparing the classical and prioritized settings, we find that the worst-case bound for label size often "translates" to a prioritized one, but also find a surprising exception to this rule.

cs.DS↗

arXiv2023

Using Deepfake Technologies for Word Emphasis Detection

In this work, we consider the task of automated emphasis detection for spoken language. This problem is challenging in that emphasis is affected by the particularities of speech of the subject, for example the subject accent, dialect or voice. To address this task, we propose to utilize deep fake technology to produce an emphasis devoid speech for this speaker. This requires extracting the text of the spoken voice, and then using a voice sample from the same speaker to produce emphasis devoid speech for this task. By comparing the generated speech with the spoken voice, we are able to isolate patterns of emphasis which are relatively easy to detect.

cs.LG↗

arXiv2022

Faster Algorithms for Orienteering and $k$-TSP

We consider the rooted orienteering problem in Euclidean space: Given $n$ points $P$ in $\mathbb R^d$, a root point $s\in P$ and a budget $\mathcal B>0$, find a path that starts from $s$, has total length at most $\mathcal B$, and visits as many points of $P$ as possible. This problem is known to be NP-hard, hence we study $(1-δ)$-approximation algorithms. The previous Polynomial-Time Approximation Scheme (PTAS) for this problem, due to Chen and Har-Peled (2008), runs in time $n^{O(d\sqrt{d}/δ)}(\log n)^{(d/δ)^{O(d)}}$, and improving on this time bound was left as an open problem. Our main contribution is a PTAS with a significantly improved time complexity of $n^{O(1/δ)}(\log n)^{(d/δ)^{O(d)}}$. A known technique for approximating the orienteering problem is to reduce it to solving $1/δ$ correlated instances of rooted $k$-TSP (a $k$-TSP tour is one that visits at least $k$ points). However, the $k$-TSP tours in this reduction must achieve a certain excess guarantee (namely, their length can surpass the optimum length only in proportion to a parameter of the optimum called excess) that is stronger than the usual $(1+δ)$-approximation. Our main technical contribution is to improve the running time of these $k$-TSP variants, particularly in its dependence on the dimension $d$. Indeed, our running time is polynomial even for a moderately large dimension, roughly up to $d=O(\log\log n)$ instead of $d=O(1)$.

cs.DS↗

arXiv2021

Learning convex polyhedra with margin

We present an improved algorithm for {\em quasi-properly} learning convex polyhedra in the realizable PAC setting from data with a margin. Our learning algorithm constructs a consistent polyhedron as an intersection of about $t \log t$ halfspaces with constant-size margins in time polynomial in $t$ (where $t$ is the number of halfspaces forming an optimal polyhedron). We also identify distinct generalizations of the notion of margin from hyperplanes to polyhedra and investigate how they relate geometrically; this result may have ramifications beyond the learning setting.

cs.LG↗

arXiv2020

Functions with average smoothness: structure, algorithms, and learning

We initiate a program of average smoothness analysis for efficiently learning real-valued functions on metric spaces. Rather than using the Lipschitz constant as the regularizer, we define a local slope at each point and gauge the function complexity as the average of these values. Since the mean can be dramatically smaller than the maximum, this complexity measure can yield considerably sharper generalization bounds -- assuming that these admit a refinement where the Lipschitz constant is replaced by our average of local slopes. Our first major contribution is to obtain just such distribution-sensitive bounds. This required overcoming a number of technical challenges, perhaps the most formidable of which was bounding the {\em empirical} covering numbers, which can be much worse-behaved than the ambient ones. Our combinatorial results are accompanied by efficient algorithms for smoothing the labels of the random sample, as well as guarantees that the extension from the sample to the whole space will continue to be, with high probability, smooth on average. Along the way we discover a surprisingly rich combinatorial and analytic structure in the function class we define.

math.ST↗

arXiv2020

Non-uniform packings

We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {\em non-uniform packing number}. We show that the non-uniform packing number can be upper-bounded in terms of the {\em average} radius of the balls, resulting in bounds of the familiar classical form.

math.MG↗

arXiv2020

Nested Barycentric Coordinate System as an Explicit Feature Map

We propose a new embedding method which is particularly well-suited for settings where the sample size greatly exceeds the ambient dimension. Our technique consists of partitioning the space into simplices and then embedding the data points into features corresponding to the simplices' barycentric coordinates. We then train a linear classifier in the rich feature space obtained from the simplices. The decision boundary may be highly non-linear, though it is linear within each simplex (and hence piecewise-linear overall). Further, our method can approximate any convex body. We give generalization bounds based on empirical margin and a novel hybrid sample compression technique. An extensive empirical evaluation shows that our method consistently outperforms a range of popular kernel embedding methods.

cs.LG↗

arXiv2020

Apportioned Margin Approach for Cost Sensitive Large Margin Classifiers

We consider the problem of cost sensitive multiclass classification, where we would like to increase the sensitivity of an important class at the expense of a less important one. We adopt an {\em apportioned margin} framework to address this problem, which enables an efficient margin shift between classes that share the same boundary. The decision boundary between all pairs of classes divides the margin between them in accordance to a given prioritization vector, which yields a tighter error bound for the important classes while also reducing the overall out-of-sample error. In addition to demonstrating an efficient implementation of our framework, we derive generalization bounds, demonstrate Fisher consistency, adapt the framework to Mercer's kernel and to neural networks, and report promising empirical results on all accounts.

cs.LG↗

arXiv2019

Classification in asymmetric spaces via sample compression

We initiate the rigorous study of classification in quasi-metric spaces. These are point sets endowed with a distance function that is non-negative and also satisfies the triangle inequality, but is asymmetric. We develop and refine a learning algorithm for quasi-metrics based on sample compression and nearest neighbor, and prove that it has favorable statistical properties.

cs.LG↗

arXiv2019

Near-linear time approximation schemes for Steiner tree and forest in low-dimensional spaces

We give an algorithm that computes a $(1+ε)$-approximate Steiner forest in near-linear time $n \cdot 2^{(1/ε)^{O(ddim^2)} (\log \log n)^2}$. This is a dramatic improvement upon the best previous result due to Chan et al., who gave a runtime of $n^{2^{O(ddim)}} \cdot 2^{(ddim/ε)^{O(ddim)} \sqrt{\log n}}$. For Steiner tree our methods achieve an even better runtime $n (\log n)^{(1/ε)^{O(ddim^2)}}$ in doubling spaces. For Euclidean space the runtime can be reduced to $2^{(1/ε)^{O(d^2)}} n \log n$, improving upon the result of Arora in fixed dimension $d$.

cs.CG↗

arXiv2018

Near-optimal sample compression for nearest neighbors

We present the first sample compression algorithm for nearest neighbors with non-trivial performance guarantees. We complement these guarantees by demonstrating almost matching hardness lower bounds, which show that our bound is nearly optimal. Our result yields new insight into margin-based nearest neighbor classification in metric spaces and allows us to significantly sharpen and simplify existing bounds. Some encouraging empirical results are also presented.

cs.LG↗

arXiv2017

Efficient Regression in Metric Spaces via Approximate Lipschitz Extension

We present a framework for performing efficient regression in general metric spaces. Roughly speaking, our regressor predicts the value at a new point by computing a Lipschitz extension --- the smoothest function consistent with the observed data --- after performing structural risk minimization to avoid overfitting. We obtain finite-sample risk bounds with minimal structural and noise assumptions, and a natural speed-precision tradeoff. The offline (learning) and online (prediction) stages can be solved by convex programming, but this naive approach has runtime complexity $O(n^3)$, which is prohibitive for large datasets. We design instead a regression algorithm whose speed and generalization performance depend on the intrinsic dimension of the data, to which the algorithm adapts. While our main innovation is algorithmic, the statistical results may also be of independent interest.

cs.LG↗

arXiv2015

Dimension reduction techniques for $\ell_p$, $1 \le p \le 2$, with applications

For Euclidean space ($\ell_2$), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss, with a host of known applications. Here, we consider the problem of dimension reduction for all $\ell_p$ spaces $1 \le p \le 2$. Although strong lower bounds are known for dimension reduction in $\ell_1$, Ostrovsky and Rabani successfully circumvented these by presenting an $\ell_1$ embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to $\ell_1$ and do not naturally extend to other norms. In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of $1 \le p \le 2$, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for $\ell_1$ can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering.

cs.CG↗

arXiv2015

Approximate nearest neighbor search for $\ell_p$-spaces ($2 < p < \infty$) via embeddings

While the problem of approximate nearest neighbor search has been well-studied for Euclidean space and $\ell_1$, few non-trivial algorithms are known for $\ell_p$ when ($2 < p < \infty$). In this paper, we revisit this fundamental problem and present approximate nearest-neighbor search algorithms which give the first non-trivial approximation factor guarantees in this setting.

cs.CG↗

arXiv2015

A light metric spanner

It has long been known that $d$-dimensional Euclidean point sets admit $(1+ε)$-stretch spanners with lightness $W_E = ε^{-O(d)}$, that is total edge weight at most $W_E$ times the weight of the minimum spaning tree of the set [DHN93]. Whether or not a similar result holds for metric spaces with low doubling dimension has remained an important open problem, and has resisted numerous attempts at resolution. In this paper, we resolve the question in the affirmative, and show that doubling spaces admit $(1+ε)$-stretch spanners with lightness $W_D = (ddim/ε)^{O(ddim)}$. Important in its own right, our result also implies a much faster polynomial-time approximation scheme for the traveling salesman problemin doubling metric spaces, improving upon the bound presented in [BGK-12].

cs.CG↗

arXiv2015

A Nonlinear Approach to Dimension Reduction

The $l_2$ flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful tool for dimension reduction. It has been conjectured that the target dimension bounds can be refined and bounded in terms of the intrinsic dimensionality of the data set (for example, the doubling dimension). One such problem was proposed by Lang and Plaut [LP01] (see also [GKL03,MatousekProblems07,ABN08,CGT10]), and is still open. We prove another result in this line of work: The snowflake metric $d^{1/2}$ of a doubling set $S \subset l_2$ embeds with constant distortion into $l_2^D$, for dimension $D$ that depends solely on the doubling constant of the metric. In fact, the distortion can be made arbitrarily close to 1, and the target dimension is polylogarithmic in the doubling constant. Our techniques are robust and extend to the more difficult spaces $l_1$ and $l_\infty$, although the dimension bounds here are quantitatively inferior than those for $l_2$.

cs.CG↗