31 research outputs found

### Improved Diversity Maximization Algorithms for Matching and Pseudoforest

In this work we consider the diversity maximization problem, where given a
data set $X$ of $n$ elements, and a parameter $k$, the goal is to pick a subset
of $X$ of size $k$ maximizing a certain diversity measure. [CH01] defined a
variety of diversity measures based on pairwise distances between the points. A
constant factor approximation algorithm was known for all those diversity
measures except ``remote-matching'', where only an $O(\log k)$ approximation
was known. In this work we present an $O(1)$ approximation for this remaining
notion. Further, we consider these notions from the perpective of composable
coresets. [IMMM14] provided composable coresets with a constant factor
approximation for all but ``remote-pseudoforest'' and ``remote-matching'',
which again they only obtained a $O(\log k)$ approximation. Here we also close
the gap up to constants and present a constant factor composable coreset
algorithm for these two notions. For remote-matching, our coreset has size only
$O(k)$, and for remote-pseudoforest, our coreset has size
$O(k^{1+\varepsilon})$ for any $\varepsilon > 0$, for an
$O(1/\varepsilon)$-approximate coreset.Comment: 27 pages, 1 table. Accepted to APPROX, 202

### Towards Tight Bounds for the Streaming Set Cover Problem

We consider the classic Set Cover problem in the data stream model. For $n$
elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a
strongly sub-linear $\tilde{O}(mn^{\delta})$ space and logarithmic
approximation factor. This yields a significant improvement over the earlier
algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of
passes. We complement this result by showing that the tradeoff between the
number of passes and space exhibited by our algorithm is tight, at least when
the approximation factor is equal to $1$. Specifically, we show that any
algorithm that computes set cover exactly using $({1 \over 2\delta}-1)$ passes
must use $\tilde{\Omega}(mn^{\delta})$ space in the regime of $m=O(n)$.
Furthermore, we consider the problem in the geometric setting where the
elements are points in $\mathbb{R}^2$ and sets are either discs, axis-parallel
rectangles, or fat triangles in the plane, and show that our algorithm (with a
slight modification) uses the optimal $\tilde{O}(n)$ space to find a
logarithmic approximation in $O(1/\delta)$ passes.
Finally, we show that any randomized one-pass algorithm that distinguishes
between covers of size 2 and 3 must use a linear (i.e., $\Omega(mn)$) amount of
space. This is the first result showing that a randomized, approximate
algorithm cannot achieve a space bound that is sublinear in the input size.
This indicates that using multiple passes might be necessary in order to
achieve sub-linear space bounds for this problem while guaranteeing small
approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201

### Approximate nearest neighbor and its many variants

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 53-55).This thesis investigates two variants of the approximate nearest neighbor problem. First, motivated by the recent research on diversity-aware search, we investigate the k-diverse near neighbor reporting problem. The problem is defined as follows: given a query point q, report the maximum diversity set S of k points in the ball of radius r around q. The diversity of a set S is measured by the minimum distance between any pair of points in S (the higher, the better). We present two approximation algorithms for the case where the points live in a d-dimensional Hamming space. Our algorithms guarantee query times that are sub-linear in n and only polynomial in the diversity parameter k, as well as the dimension d. For low values of k, our algorithms achieve sub-linear query times even if the number of points within distance r from a query q is linear in n. To the best of our knowledge, these are the first known algorithms of this type that offer provable guarantees. In the other variant, we consider the approximate line near neighbor (LNN) problem. Here, the database consists of a set of lines instead of points but the query is still a point. Let L be a set of n lines in the d dimensional euclidean space Rd. The goal is to preprocess the set of lines so that we can answer the Line Near Neighbor (LNN) queries in sub-linear time. That is, given the query point ... we want to report a line ... (if there is any), such that ... for some threshold value r, where ... is the euclidean distance between them. We start by illustrating the solution to the problem in the case where there are only two lines in the database and present a data structure in this case. Then we show a recursive algorithm that merges these data structures and solve the problem for the general case of n lines. The algorithm has polynomial space and performs only a logarithmic number of calls to the approximate nearest neighbor subproblem.by Sepideh Mahabadi.S.M

### Approximate Sparse Linear Regression

In the Sparse Linear Regression (SLR) problem, given a d x n matrix M and a d-dimensional query q, the goal is to compute a k-sparse n-dimensional vector tau such that the error ||M tau - q|| is minimized. This problem is equivalent to the following geometric problem: given a set P of n points and a query point q in d dimensions, find the closest k-dimensional subspace to q, that is spanned by a subset of k points in P. In this paper, we present data-structures/algorithms and conditional lower bounds for several variants of this problem (such as finding the closest induced k dimensional flat/simplex instead of a subspace).
In particular, we present approximation algorithms for the online variants of the above problems with query time O~(n^{k-1}), which are of interest in the "low sparsity regime" where k is small, e.g., 2 or 3. For k=d, this matches, up to polylogarithmic factors, the lower bound that relies on the affinely degenerate conjecture (i.e., deciding if n points in R^d contains d+1 points contained in a hyperplane takes Omega(n^d) time). Moreover, our algorithms involve formulating and solving several geometric subproblems, which we believe to be of independent interest

### Approximation Algorithms for Fair Range Clustering

This paper studies the fair range clustering problem in which the data points
are from different demographic groups and the goal is to pick $k$ centers with
the minimum clustering cost such that each group is at least minimally
represented in the centers set and no group dominates the centers set. More
precisely, given a set of $n$ points in a metric space $(P,d)$ where each point
belongs to one of the $\ell$ different demographics (i.e., $P = P_1 \uplus P_2
\uplus \cdots \uplus P_\ell$) and a set of $\ell$ intervals $[\alpha_1,
\beta_1], \cdots, [\alpha_\ell, \beta_\ell]$ on desired number of centers from
each group, the goal is to pick a set of $k$ centers $C$ with minimum
$\ell_p$-clustering cost (i.e., $(\sum_{v\in P} d(v,C)^p)^{1/p}$) such that for
each group $i\in \ell$, $|C\cap P_i| \in [\alpha_i, \beta_i]$. In particular,
the fair range $\ell_p$-clustering captures fair range $k$-center, $k$-median
and $k$-means as its special cases. In this work, we provide efficient constant
factor approximation algorithms for fair range $\ell_p$-clustering for all
values of $p\in [1,\infty)$.Comment: ICML 202

### Adaptive Sketches for Robust Regression with Importance Sampling

We introduce data structures for solving robust regression through stochastic gradient descent (SGD) by sampling gradients with probability proportional to their norm, i.e., importance sampling. Although SGD is widely used for large scale machine learning, it is well-known for possibly experiencing slow convergence rates due to the high variance from uniform sampling. On the other hand, importance sampling can significantly decrease the variance but is usually difficult to implement because computing the sampling probabilities requires additional passes over the data, in which case standard gradient descent (GD) could be used instead. In this paper, we introduce an algorithm that approximately samples T gradients of dimension d from nearly the optimal importance sampling distribution for a robust regression problem over n rows. Thus our algorithm effectively runs T steps of SGD with importance sampling while using sublinear space and just making a single pass over the data. Our techniques also extend to performing importance sampling for second-order optimization