Searcharxiv⌕ Search

arXiv subjects

Ragesh Jaiswal

Publications and source records attributed to Ragesh Jaiswal.

29 records · Page 2Linked to original sources
arXiv2019

A note on the relation between XOR and Selective XOR Lemmas

Given an unpredictable Boolean function $f: \{0, 1\}^n \rightarrow \{0, 1\}$, the standard Yao's XOR lemma is a statement about the unpredictability of computing $\oplus_{i \in [k]}f(x_i)$ given $x_1, ..., x_k \in \{0, 1\}^n$, whereas the Selective XOR lemma is a statement about the unpredictability of computing $\oplus_{i \in S}f(x_i)$ given $x_1, ..., x_k \in \{0, 1\}^n$ and $S \subseteq \{1, ..., k\}$. We give a reduction from the Selective XOR lemma to the standard XOR lemma. Our reduction gives better quantitative bounds for certain choice of parameters and does not require the assumption of being able to sample $(x, f(x))$ pairs.

cs.CC↗
arXiv2017

Approximate Correlation Clustering Using Same-Cluster Queries

Ashtiani et al. (NIPS 2016) introduced a semi-supervised framework for clustering (SSAC) where a learner is allowed to make same-cluster queries. More specifically, in their model, there is a query oracle that answers queries of the form given any two vertices, do they belong to the same optimal cluster?. Ashtiani et al. showed the usefulness of such a query framework by giving a polynomial time algorithm for the k-means clustering problem where the input dataset satisfies some separation condition. Ailon et al. extended the above work to the approximation setting by giving an efficient (1+\eps)-approximation algorithm for k-means for any small \eps > 0 and any dataset within the SSAC framework. In this work, we extend this line of study to the correlation clustering problem. Correlation clustering is a graph clustering problem where pairwise similarity (or dissimilarity) information is given for every pair of vertices and the objective is to partition the vertices into clusters that minimise the disagreement (or maximises agreement) with the pairwise information given as input. These problems are popularly known as MinDisAgree and MaxAgree problems, and MinDisAgree[k] and MaxAgree[k] are versions of these problems where the number of optimal clusters is at most k. There exist Polynomial Time Approximation Schemes (PTAS) for MinDisAgree[k] and MaxAgree[k] where the approximation guarantee is (1+\eps) for any small \eps and the running time is polynomial in the input parameters but exponential in k and 1/\eps. We obtain an (1+\eps)-approximation algorithm for any small \eps with running time that is polynomial in the input parameters and also in k and 1/\eps. We also give non-trivial upper and lower bounds on the number of same-cluster queries, the lower bound being based on the Exponential Time Hypothesis (ETH).

cs.DS↗
arXiv2017

Approximate Clustering with Same-Cluster Queries

Ashtiani et al. proposed a Semi-Supervised Active Clustering framework (SSAC), where the learner is allowed to make adaptive queries to a domain expert. The queries are of the kind "do two given points belong to the same optimal cluster?" There are many clustering contexts where such same-cluster queries are feasible. Ashtiani et al. exhibited the power of such queries by showing that any instance of the $k$-means clustering problem, with additional margin assumption, can be solved efficiently if one is allowed $O(k^2 \log{k} + k \log{n})$ same-cluster queries. This is interesting since the $k$-means problem, even with the margin assumption, is $\mathsf{NP}$-hard. In this paper, we extend the work of Ashtiani et al. to the approximation setting showing that a few of such same-cluster queries enables one to get a polynomial-time $(1 + \varepsilon)$-approximation algorithm for the $k$-means problem without any margin assumption on the input dataset. Again, this is interesting since the $k$-means problem is $\mathsf{NP}$-hard to approximate within a factor $(1 + c)$ for a fixed constant $0 < c < 1$. The number of same-cluster queries used is $\textrm{poly}(k/\varepsilon)$ which is independent of the size $n$ of the dataset. Our algorithm is based on the $D^2$-sampling technique. We also give a conditional lower bound on the number of same-cluster queries showing that if the Exponential Time Hypothesis (ETH) holds, then any such efficient query algorithm needs to make $Ω\left(\frac{k}{poly \log k} \right)$ same-cluster queries. Our algorithm can be extended for the case when the oracle is faulty. Another result we show with respect to the $k$-means++ seeding algorithm is that a small modification to the $k$-means++ seeding algorithm within the SSAC framework converts it to a constant factor approximation algorithm instead of the well known $O(\log{k})$-approximation algorithm.

cs.DS↗
arXiv2015

Faster Algorithms for the Constrained k-means Problem

The classical center based clustering problems such as $k$-means/median/center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise in machine learning where the optimal clusters do not follow such a locality property. Consider a variant of the $k$-means problem that may be regarded as a general version of such problems. Here, the optimal clusters $O_1, ..., O_k$ are an arbitrary partition of the dataset and the goal is to output $k$-centers $c_1, ..., c_k$ such that the objective function $\sum_{i=1}^{k} \sum_{x \in O_{i}} ||x - c_{i}||^2$ is minimized. It is not difficult to argue that any algorithm (without knowing the optimal clusters) that outputs a single set of $k$ centers, will not behave well as far as optimizing the above objective function is concerned. However, this does not rule out the existence of algorithms that output a list of such $k$ centers such that at least one of these $k$ centers behaves well. Given an error parameter $\varepsilon > 0$, let $\ell$ denote the size of the smallest list of $k$-centers such that at least one of the $k$-centers gives a $(1+\varepsilon)$ approximation w.r.t. the objective function above. In this paper, we show an upper bound on $\ell$ by giving a randomized algorithm that outputs a list of $2^{\tilde{O}(k/\varepsilon)}$ $k$-centers. We also give a closely matching lower bound of $2^{\tildeΩ(k/\sqrt{\varepsilon})}$. Moreover, our algorithm runs in time $O \left(n d \cdot 2^{\tilde{O}(k/\varepsilon)} \right)$. This is a significant improvement over the previous result of Ding and Xu who gave an algorithm with running time $O \left(n d \cdot (\log{n})^{k} \cdot 2^{poly(k/\varepsilon)} \right)$ and output a list of size $O \left((\log{n})^k \cdot 2^{poly(k/\varepsilon)} \right)$.

cs.DS↗
arXiv2015

Sampling in Space Restricted Settings

Space efficient algorithms play a central role in dealing with large amount of data. In such settings, one would like to analyse the large data using small amount of "working space". One of the key steps in many algorithms for analysing large data is to maintain a (or a small number) random sample from the data points. In this paper, we consider two space restricted settings -- (i) streaming model, where data arrives over time and one can use only a small amount of storage, and (ii) query model, where we can structure the data in low space and answer sampling queries. In this paper, we prove the following results in above two settings: - In the streaming setting, we would like to maintain a random sample from the elements seen so far. We prove that one can maintain a random sample using $O(\log n)$ random bits and $O(\log n)$ space, where $n$ is the number of elements seen so far. We can extend this to the case when elements have weights as well. - In the query model, there are $n$ elements with weights $w_1, ..., w_n$ (which are $w$-bit integers) and one would like to sample a random element with probability proportional to its weight. Bringmann and Larsen (STOC 2013) showed how to sample such an element using $nw +1 $ space (whereas, the information theoretic lower bound is $n w$). We consider the approximate sampling problem, where we are given an error parameter $\varepsilon$, and the sampling probability of an element can be off by an $\varepsilon$ factor. We give matching upper and lower bounds for this problem.

cs.DS↗
arXiv2014

A tight lower bound instance for k-means++ in constant dimension

The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the first center randomly from the given points. For $i > 1$, pick a point to be the $i^{th}$ center with probability proportional to the square of the Euclidean distance of this point to the closest previously $(i-1)$ chosen centers. The k-means++ seeding algorithm is not only simple and fast but also gives an $O(\log{k})$ approximation in expectation as shown by Arthur and Vassilvitskii. There are datasets on which this seeding algorithm gives an approximation factor of $Ω(\log{k})$ in expectation. However, it is not clear from these results if the algorithm achieves good approximation factor with reasonably high probability (say $1/poly(k)$). Brunsch and Röglin gave a dataset where the k-means++ seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability that is exponentially small in $k$. However, this and all other known lower-bound examples are high dimensional. So, an open problem was to understand the behavior of the algorithm on low dimensional datasets. In this work, we give a simple two dimensional dataset on which the seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability exponentially small in $k$. This solves open problems posed by Mahajan et al. and by Brunsch and Röglin.

cs.DS↗
arXiv2013

A bad 2-dimensional instance for k-means++

The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: {quote} Pick the first center randomly from among the given points. For $i > 1$, pick a point to be the $i^{th}$ center with probability proportional to the square of the Euclidean distance of this point to the previously $(i-1)$ chosen centers. {quote} The k-means++ seeding algorithm is not only simple and fast but gives an $O(\log{k})$ approximation in expectation as shown by Arthur and Vassilvitskii \cite{av07}. There are datasets \cite{av07,adk09} on which this seeding algorithm gives an approximation factor $Ω(\log{k})$ in expectation. However, it is not clear from these results if the algorithm achieves good approximation factor with reasonably large probability (say $1/poly(k)$). Brunsch and Röglin \cite{br11} gave a dataset where the k-means++ seeding algorithm achieves an approximation ratio of $(2/3 - ε)\cdot \log{k}$ only with probability that is exponentially small in $k$. However, this and all other known {\em lower-bound examples} \cite{av07,adk09} are high dimensional. So, an open problem is to understand the behavior of the algorithm on low dimensional datasets. In this work, we give a simple two dimensional dataset on which the seeding algorithm achieves an approximation ratio $c$ (for some universal constant $c$) only with probability exponentially small in $k$. This is the first step towards solving open problems posed by Mahajan et al \cite{mnv12} and by Brunsch and Röglin \cite{br11}.

cs.DS↗
arXiv2012

On the Distribution of the Fourier Spectrum of Halfspaces

Bourgain showed that any noise stable Boolean function $f$ can be well-approximated by a junta. In this note we give an exponential sharpening of the parameters of Bourgain's result under the additional assumption that $f$ is a halfspace.

cs.CC↗
arXiv2012

A simple D^2-sampling based PTAS for k-means and other Clustering Problems

Given a set of points $P \subset \mathbb{R}^d$, the $k$-means clustering problem is to find a set of $k$ {\em centers} $C = \{c_1,...,c_k\}, c_i \in \mathbb{R}^d,$ such that the objective function $\sum_{x \in P} d(x,C)^2$, where $d(x,C)$ denotes the distance between $x$ and the closest center in $C$, is minimized. This is one of the most prominent objective functions that have been studied with respect to clustering. $D^2$-sampling \cite{ArthurV07} is a simple non-uniform sampling technique for choosing points from a set of points. It works as follows: given a set of points $P \subseteq \mathbb{R}^d$, the first point is chosen uniformly at random from $P$. Subsequently, a point from $P$ is chosen as the next sample with probability proportional to the square of the distance of this point to the nearest previously sampled points. $D^2$-sampling has been shown to have nice properties with respect to the $k$-means clustering problem. Arthur and Vassilvitskii \cite{ArthurV07} show that $k$ points chosen as centers from $P$ using $D^2$-sampling gives an $O(\log{k})$ approximation in expectation. Ailon et. al. \cite{AJMonteleoni09} and Aggarwal et. al. \cite{AggarwalDK09} extended results of \cite{ArthurV07} to show that $O(k)$ points chosen as centers using $D^2$-sampling give $O(1)$ approximation to the $k$-means objective function with high probability. In this paper, we further demonstrate the power of $D^2$-sampling by giving a simple randomized $(1 + ε)$-approximation algorithm that uses the $D^2$-sampling in its core.

cs.DS↗
arXiv2009

Bounded Independence Fools Halfspaces

We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error \eps for k = O(\log^2(1/\eps) /\eps^2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and Peled (2007) showing that k = Ω(1/(\eps^2 \cdot \log(1/\eps))). Using standard constructions of k-wise independent distributions, we obtain the first explicit pseudorandom generators G: {-1,1}^s --> {-1,1}^n that fool halfspaces. Specifically, we fool halfspaces with error eps and seed length s = k \log n = O(\log n \cdot \log^2(1/\eps) /\eps^2). Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Computational Complexity 2007).

cs.CC↗