Approximation algorithms for geometric dispersion

The most basic form of the max-sum dispersion problem (MSD) is as follows: given n points in R^q and an integer k, select a set of k points such that the sum of the pairwise distances within the set is maximal. This is a prominent diversity problem, with wide applications in web search and information retrieval, where one needs to find a small and diverse representative subset of a large dataset. The problem has recently received a great deal of attention in the computational geometry and operations research communities; and since it is NP-hard, research has focused on efficient heuristics and approximation algorithms. Several classes of distance functions have been considered in the literature. Many of the most common distances used in applications are induced by a norm in a real vector space. The focus of this thesis is on MSD over these geometric instances. We provide for it simple and fast polynomial-time approximation schemes (PTASs), as well as improved constant-factor approximation algorithms. We pay special attention to the class of negative-type distances, a class that includes Euclidean and Manhattan distances, among many others. In order to exploit the properties of this class, we apply several techniques and results from the theory of isometric embeddings. We explore the following variations of the MSD problem: matroid and matroid-intersection constraints, knapsack constraints, and the mixed-objective problem that maximizes a combination of the sum of pairwise distances with a submodular monotone function. In addition to approximation algorithms, we present a core-set for geometric instances of low dimension, and we discuss the efficient implementation of some of our algorithms for massive datasets, using the streaming and distributed models of computation.

Subdeterminant Maximization via Nonconvex Relaxations and Anti-Concentration

Javad Ebrahimi Boroojeni, Damian Mateusz Straszak, Nisheeth Vishnoi

Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors v(1), ..., v(m) is an element of R-d and a constraint family B subset of 2([m]), find a set S. B that maximizes the squared volume of the simplex spanned by the vectors in S. A motivating example is the ubiquitous data-summarization problem in machine learning and information retrieval where one is given a collection of feature vectors that represent data such as documents or images. The volume of a collection of vectors is used as a measure of their diversity, and partition or matroid constraints over [m] are imposed in order to ensure resource or fairness constraints. Even with a simple cardinality constraint (B = (([m])(r))), the problem becomes NP-hard and has received much attention starting with a result by Khachiyan [1] who gave an r(O(r)) approximation algorithm for this problem. Recently, Nikolov and Singh [2] presented a convex program and showed how it can be used to estimate the value of the most diverse set when there are multiple cardinality constraints (i.e., when B corresponds to a partition matroid). Their proof of the integrality gap of the convex program relied on an inequality by Gurvits [3], and was recently extended to regular matroids [4], [5]. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms - that also output a set - remained open. The main contribution of this paper is to give the first approximation algorithms for both partition and regular matroids. We present novel formulations for the subdeterminant maximization problem for these matroids; this reduces them to the problem of finding a point that maximizes the absolute value of a nonconvex function over a Cartesian product of probability simplices. The technical core of our results is a new anti-concentration inequality for dependent random variables that arise from these functions which allows us to relate the optimal value of these nonconvex functions to their value at a random point. Unlike prior work on the constrained subdeterminant maximization problem, our proofs do not rely on real-stability or convexity and could be of independent interest both in algorithms and complexity where anti-concentration phenomena has recently been deployed.

Ieee2017

Realtime Face Tracking and Animation

Sofien Bouaziz

Capturing and processing human geometry, appearance, and motion is at the core of computer graphics, computer vision, and human-computer interaction. The high complexity of human geometry and motion dynamics, and the high sensitivity of the human visual system to variations and subtleties in faces and bodies make the 3D acquisition and reconstruction of humans in motion a challenging task. Digital humans are often created through a combination of 3D scanning, appearance acquisition, and motion capture, leading to stunning results in recent feature films. However, these methods typically require complex acquisition systems and substantial manual post-processing. As a result, creating and animating high-quality digital avatars entails long turn-around times and substantial production costs. Recent technological advances in RGB-D devices, such as Microsoft Kinect, brought new hopes for realtime, portable, and affordable systems allowing to capture facial expressions as well as hand and body motions. RGB-D devices typically capture an image and a depth map. This permits to formulate the motion tracking problem as a 2D/3D non-rigid registration of a deformable model to the input data. We introduce a novel face tracking algorithm that combines geometry and texture registration with pre-recorded animation priors in a single optimization. This led to unprecedented face tracking quality on a low cost consumer level device. The main drawback of this approach in the context of consumer applications is the need for an offline user-specific training. Robust and efficient tracking is achieved by building an accurate 3D expression model of the user's face who is scanned in a predefined set of facial expressions. We extended this approach removing the need of a user-specific training or calibration, or any other form of manual assistance, by modeling online a 3D user-specific dynamic face model. In complement of a realtime face tracking and modeling algorithm, we developed a novel system for animation retargeting that allows learning a high-quality mapping between motion capture data and arbitrary target characters. We addressed one of the main challenges of existing example-based retargeting methods, the need for a large number of accurate training examples to define the correspondence between source and target expression spaces. We showed that this number can be significantly reduced by leveraging the information contained in unlabeled data, i.e. facial expressions in the source or target space without corresponding poses. Finally, we present a novel realtime physics-based animation technique allowing to simulate a large range of deformable materials such as fat, flesh, hair, or muscles. This approach could be used to produce more lifelike animations by enhancing the animated avatars with secondary effects. We believe that the realtime face tracking and animation pipeline presented in this thesis has the potential to inspire numerous future research in the area of computer-generated animation. Already, several ideas presented in thesis have been successfully used in industry and this work gave birth to the startup company faceshift AG.

EPFL2015

Subdeterminant Maximization via Nonconvex Relaxations and Anti-Concentration

Javad Ebrahimi Boroojeni, Damian Mateusz Straszak, Nisheeth Vishnoi

Ieee2017

Realtime Face Tracking and Animation

Sofien Bouaziz

EPFL2015