arXiv subjects

Mehrdad Ghadiri

Publications and source records attributed to Mehrdad Ghadiri.

26 records · Page 2Linked to original sources

arXiv2020

Socially Fair k-Means Clustering

We show that the popular k-means clustering algorithm (Lloyd's heuristic), used for a variety of scientific data, can result in outcomes that are unfavorable to subgroups of data (e.g., demographic groups). Such biased clusterings can have deleterious implications for human-centric applications such as resource allocation. We present a fair k-means objective and algorithm to choose cluster centers that provide equitable costs for different groups. The algorithm, Fair-Lloyd, is a modification of Lloyd's heuristic for k-means, inheriting its simplicity, efficiency, and stability. In comparison with standard Lloyd's, we find that on benchmark datasets, Fair-Lloyd exhibits unbiased performance by ensuring that all groups have equal costs in the output k-clustering, while incurring a negligible increase in running time, thus making it a viable fair option wherever k-means is currently used.

cs.LG↗

arXiv2020

A Parameterized Family of Meta-Submodular Functions

Submodular function maximization has found a wealth of new applications in machine learning models during the past years. The related supermodular maximization models (submodular minimization) also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints. Hence, while there are well-developed tools for maximizing a submodular function subject to a matroid constraint, there is much less work on the corresponding supermodular maximization problems. We give a broad parameterized family of monotone functions which includes submodular functions and a class of supermodular functions containing diversity functions. Functions in this parameterized family are called \emph{$γ$-meta-submodular}. We develop local search algorithms with approximation factors that depend only on the parameter $γ$. We show that the $γ$-meta-submodular families include well-known classes of functions such as meta-submodular functions ($γ=0$), metric diversity functions and proportionally submodular functions (both with $γ=1$), diversity functions based on negative-type distances or Jensen-Shannon divergence (both with $γ=2$), and $σ$-semi metric diversity functions ($γ= σ$).

cs.DS↗

arXiv2020

Beyond Submodular Maximization via One-Sided Smoothness

The multilinear framework has achieved the breakthrough $1-1/e$ approximation for maximizing a monotone submodular function subject to a matroid constraint. This framework has a continuous optimization part and a rounding part. We extend both parts to a wider array of problems. In particular, we make a conceptual contribution by identifying a family of parameterized functions. As a running example we focus on solving diversity problems $\max f(S)=\frac{1}{2}\sum_{i,j\in A}A_{ij}:S\in\mathcal{M}$, where $\mathcal{M}$ is a matroid. These diversity functions have $A_{ij}\geq 0$ as a measure of dissimilarity of $i,j$, and $A$ has $0$-diagonal. The multilinear framework cannot be directly applied to the multilinear extension of such functions. We introduce a new parameter for functions $F\in{\bf C}^2$ which measures the approximability of the associated problem $\max\{F(x):x\in P\}$, for solvable downwards-closed polytopes $P$. A function $F$ is called one-sided $σ$-smooth if $\frac{1}{2}u^T\nabla^2 F(x) u\leqσ\cdot\frac{||u||_1}{||x||_1}u^T\nabla F(x)$ for all $u,x\geq 0$, $x\neq 0$. We give an $Ω(1/σ)$-approximation for the maximization problem of monotone, normalized one-sided $σ$-smooth $F$ with an additional property: non-positive third order partial derivatives. Using the multilinear framework and new matroid rounding techniques for quadratic objectives, we give an $Ω(1/σ^{3/2})$-approximation for maximizing a $σ$-semi-metric diversity function subject to matroid constraint. This improves upon the previous best bound of $Ω(1/σ^2)$ and we give evidence that it may be tight. For general one-sided smooth functions, we show the continuous process gives an $Ω(1/3^{2σ})$-approximation, independent of $n$. In this setting, by discretizing, we present a poly-time algorithm for multilinear one-sided $σ$-smooth functions.

cs.DS↗

arXiv2019

Distributed Maximization of Submodular plus Diversity Functions for Multi-label Feature Selection on Huge Datasets

There are many problems in machine learning and data mining which are equivalent to selecting a non-redundant, high "quality" set of objects. Recommender systems, feature selection, and data summarization are among many applications of this. In this paper, we consider this problem as an optimization problem that seeks to maximize the sum of a sum-sum diversity function and a non-negative monotone submodular function. The diversity function addresses the redundancy, and the submodular function controls the predictive quality. We consider the problem in big data settings (in other words, distributed and streaming settings) where the data cannot be stored on a single machine or the process time is too high for a single machine. We show that a greedy algorithm achieves a constant factor approximation of the optimal solution in these settings. Moreover, we formulate the multi-label feature selection problem as such an optimization problem. This formulation combined with our algorithm leads to the first distributed multi-label feature selection method. We compare the performance of this method with centralized multi-label feature selection methods in the literature, and we show that its performance is comparable or in some cases is even better than current centralized multi-label feature selection methods.

cs.LG↗

arXiv2016

Minimizing the Total Movement for Movement to Independence Problem on a Line

Given a positive real value $δ$, a set $P$ of points along a line and a distance function $d$, in the movement to independence problem, we wish to move the points to new positions on the line such that for every two points $p_{i},p_{j} \in P$, we have $d(p_{i},p_{j}) \geq δ$ while minimizing the sum of movements of all points. This measure of the cost for moving the points was previously unsolved in this setting. However for different cost measures there are algorithms of $O(n \log(n))$ or of $O(n)$. We present an $O(n \log(n))$ algorithm for the points on a line and thus conclude the setting in one dimension.

cs.CG↗

arXiv2015

Active Distance-Based Clustering using K-medoids

k-medoids algorithm is a partitional, centroid-based clustering algorithm which uses pairwise distances of data points and tries to directly decompose the dataset with $n$ points into a set of $k$ disjoint clusters. However, k-medoids itself requires all distances between data points that are not so easy to get in many applications. In this paper, we introduce a new method which requires only a small proportion of the whole set of distances and makes an effort to estimate an upper-bound for unknown distances using the inquired ones. This algorithm makes use of the triangle inequality to calculate an upper-bound estimation of the unknown distances. Our method is built upon a recursive approach to cluster objects and to choose some points actively from each bunch of data and acquire the distances between these prominent points from oracle. Experimental results show that the proposed method using only a small subset of the distances can find proper clustering on many real-world and synthetic datasets.

cs.LG↗

arXiv2015

Even and Odd Cycles Passing a Given Edge or a Vertex

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in $2$-connected or $2$-edge connected graphs. We provide some similar conditions for the existence of an odd or even circuit that passing a given vertex or an edge in 2-edge connected graphs. We show that if $G$ is a $2$-connected $k$-regular graph, $k \geq 3$, then every edge of $G$ is contained in an even cycle. We also prove that in a $2$-edge connected graph, if a vertex has odd degree, then there is an even cycle containing this vertex.

cs.DM↗

arXiv2015

Max-Sum Diversification, Monotone Submodular Functions and Semi-metric Spaces

In many applications such as web-based search, document summarization, facility location and other applications, the results are preferable to be both representative and diversified subsets of documents. The goal of this study is to select a good "quality", bounded-size subset of a given set of items, while maintaining their diversity relative to a semi-metric distance function. This problem was first studied by Borodin et al\cite{borodin}, but a crucial property used throughout their proof is the triangle inequality. In this modified proof, we want to relax the triangle inequality and relate the approximation ratio of max-sum diversification problem to the parameter of the relaxed triangle inequality in the normal form of the problem (i.e., a uniform matroid) and also in an arbitrary matroid.

cs.LG↗