Parameterized complexity of PCA

We discuss some recent progress in the study of Principal Component Analysis (PCA) from the perspective of Parameterized Complexity.publishedVersio

Analysis of nonlinear modes of variation for functional data

A set of curves or images of similar shape is an increasingly common functional data set collected in the sciences. Principal Component Analysis (PCA) is the most widely used technique to decompose variation in functional data. However, the linear modes of variation found by PCA are not always interpretable by the experimenters. In addition, the modes of variation of interest to the experimenter are not always linear. We present in this paper a new analysis of variance for Functional Data. Our method was motivated by decomposing the variation in the data into predetermined and interpretable directions (i.e. modes) of interest. Since some of these modes could be nonlinear, we develop a new defined ratio of sums of squares which takes into account the curvature of the space of variation. We discuss, in the general case, consistency of our estimates of variation, using mathematical tools from differential geometry and shape statistics. We successfully applied our method to a motivating example of biological data. This decomposition allows biologists to compare the prevalence of different genetic tradeoffs in a population and to quantify the effect of selection on evolution.Comment: Published in at http://dx.doi.org/10.1214/07-EJS080 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Verification of parameterized communicating automata via split-width

International audienceWe study verification problems for distributed systems communicating via unbounded FIFO channels. The number of processes of the system as well as the communication topology are not fixed a priori. Systems are given by parameterized communicating automata (PCAs) which can be run on any communication topology of bounded degree, with arbitrarily many processes. Such systems are Turing powerful so we concentrate on under-approximate verification. We extend the notion of split-width to behaviors of PCAs. We show that emptiness, reachability and model-checking problems of PCAs are decidable when restricted to behaviors of bounded split-width. Reachability and emptiness are Exptime-complete, but only polynomial in the size of the PCA. We also describe several concrete classes of bounded split-width, for which we prove similar results

The phenomenological approach to modeling the dark energy

In this mini-review we discuss first why we should investigate cosmological models beyond LCDM. We then show how to describe dark energy or modified gravity models in a fluid language with the help of one background and two perturbation quantities. We review a range of dark energy models and study how they fit into the phenomenological framework, including generalizations like phantom crossing, sound speeds different from c and non-zero anisotropic stress, and how these effective quantities are linked to the underlying physical models. We also discuss the limits of what can be measured with cosmological data, and some challenges for the framework.Comment: 44 pages, 5 figures; accepted review article to appear in a special volume of the "Comptes Rendus de l'Academie des Sciences" about Dark Energy and Dark Matte

PCPs and Instance Compression from a Cryptographic Lens

Modern cryptography fundamentally relies on the assumption that the adversary trying to break the scheme is computationally bounded. This assumption lets us construct cryptographic protocols and primitives that are known to be impossible otherwise. In this work we explore the effect of bounding the adversary\u27s power in other information theoretic proof-systems and show how to use this assumption to bypass impossibility results. We first consider the question of constructing succinct PCPs. These are PCPs whose length is polynomial only in the length of the original NP witness (in contrast to standard PCPs whose length is proportional to the non-deterministic verification time). Unfortunately, succinct PCPs are known to be impossible to construct under standard complexity assumptions. Assuming the sub-exponential hardness of the learning with errors (LWE) problem, we construct succinct probabilistically checkable arguments or PCAs (Zimand 2001, Kalai and Raz 2009), which are PCPs in which soundness is guaranteed against efficiently generated false proofs. Our PCA construction is for every NP relation that can be verified by a small-depth circuit (e.g., SAT, clique, TSP, etc.) and in contrast to prior work is publicly verifiable and has constant query complexity. Curiously, we also show, as a proof-of-concept, that such publicly-verifiable PCAs can be used to derive hardness of approximation results. Second, we consider the notion of Instance Compression (Harnik and Naor, 2006). An instance compression scheme lets one compress, for example, a CNF formula $\varphi$ on $m$ variables and $n \gg m$ clauses to a new formula $\varphi\u27$ with only $poly(m)$ clauses, so that $\varphi$ is satisfiable if and only if $\varphi\u27$ is satisfiable. Instance compression has been shown to be closely related to succinct PCPs and is similarly highly unlikely to exist. We introduce a computational analog of instance compression in which we require that if $\varphi$ is unsatisfiable then $\varphi\u27$ is effectively unsatisfiable, in the sense that it is computationally infeasible to find a satisfying assignment for $\varphi\u27$ (although such an assignment may exist). Assuming the same sub-exponential LWE assumption, we construct such computational instance compression schemes for every bounded-depth NP relation. As an application, this lets one compress $k$ formulas $\phi_1,\dots,\phi_k$ into a single short formula $\phi$ that is effectively satisfiable if and only if at least one of the original formulas was satisfiable

Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness

Understanding the singular value spectrum of an n x n matrix A is a fundamental task in countless numerical computation and data analysis applications. In matrix multiplication time, it is possible to perform a full SVD of A and directly compute the singular values sigma_1,...,sigma_n. However, little is known about algorithms that break this runtime barrier. Using tools from stochastic trace estimation, polynomial approximation, and fast linear system solvers, we show how to efficiently isolate different ranges of A\u27s spectrum and approximate the number of singular values in these ranges. We thus effectively compute an approximate histogram of the spectrum, which can stand in for the true singular values in many applications. We use our histogram primitive to give the first algorithms for approximating a wide class of symmetric matrix norms and spectral sums faster than the best known runtime for matrix multiplication. For example, we show how to obtain a (1 + epsilon) approximation to the Schatten 1-norm (i.e. the nuclear or trace norm) in just ~ O((nnz(A)n^{1/3} + n^2)epsilon^{-3}) time for A with uniform row sparsity or tilde O(n^{2.18} epsilon^{-3}) time for dense matrices. The runtime scales smoothly for general Schatten-p norms, notably becoming tilde O (p nnz(A) epsilon^{-3}) for any real p >= 2. At the same time, we show that the complexity of spectrum approximation is inherently tied to fast matrix multiplication in the small epsilon regime. We use fine-grained complexity to give conditional lower bounds for spectrum approximation, showing that achieving milder epsilon dependencies in our algorithms would imply triangle detection algorithms for general graphs running in faster than state of the art matrix multiplication time. This further implies, through a reduction of (Williams & William, 2010), that highly accurate spectrum approximation algorithms running in subcubic time can be used to give subcubic time matrix multiplication. As an application of our bounds, we show that precisely computing all effective resistances in a graph in less than matrix multiplication time is likely difficult, barring a major algorithmic breakthrough

Visualization of state transition graphs

State transition graphs are important in computer science and engineering where they are used to analyze the behavior of computer-based systems. In such a graph nodes represent states a system can be in. Links, or directed edges, represent transitions between states. Research in visualization investigates the application of interactive computer graphics to understand large and complex data sets. Large state transition graphs fall into this category. They often contain tens of thousands of nodes, or more, and tens to hundreds of thousands of edges. Also, they describe system behavior at a low abstraction level. This hinders analysis and insight. This dissertation presents a number of techniques for the interactive visualization of state transition graphs. Much of the work takes advantage of multivariate data associated with nodes and edges. Using an experimental approach, several new methods were developed in close collaboration with a number of users. The following approaches were pursued: • Selection and projection. This technique provides the user with visual support to select a subset of node attributes. Consequently, the state transition graph is projected to 2D and visualized in a second, correlated visualization. • Attribute-based clustering. By specifying subsets of node attributes and clustering based on these, the user generates simplified abstractions of a state transition graph. Clustering generates hierarchical, relational, and metric data, which are represented in a single visualization. • User-defined diagrams. With this technique the user investigates state transition graphs with custom diagrams. Diagrams are parameterized by linking their graphical properties to the data. Diagrams are integrated in a number of correlated visualizations. • Multiple views on traces. System traces are linear paths in state transition graphs. This technique provides the user with different perspectives on traces. • Querying nodes and edges. Direct manipulation enables the user to interactively inspect and query state transition graphs. In this way relations and patterns can be investigated based on data associated with nodes and edges. This dissertation shows that interactive visualization can play a role during the analysis of state transition graphs. The ability to interrogate visual representations of such graphs allows users to enhance their knowledge of the modeled systems. It is shown how the above techniques enable users to answer questions about their data. A number of case studies, developed in collaboration with system analysts, are presented. Finally, solutions to challenges encountered during the development of the visualization techniques are discussed. Insights generic to the field of visualization are considered and directions for future work are recommended

2D and 3D digital shape modelling strategies

Image segmentation of organs in medical images using model-based approaches requires a priori information which is often given by manually tagging landmarks on a training set of shapes. This is a tedious, time-consuming, and error prone task. To overcome some of these drawbacks, several automatic methods were devised. Identification of the same homologous set of points in a training set of object shapes is the most crucial step in Active Shape Modelling, which has encountered several challenges. The most crucial among these are: (C1) defining and characterizing landmarks; (C2) obtaining landmarks at the desired level of detail; (C3) ensuring homology; (C4) generalizing to n>2 dimensions; (C5) achieving practical computations. This thesis proposes several novel modelling techniques attempting to meet C1-C5. In this process, this thesis makes the following key contributions: the concept of local scale for shapes; the idea of allowing level of detail for selecting landmarks; the concept of equalization of shape variance for selecting landmarks; the idea of recursively subdividing shapes and letting the sub-shapes guide landmark selection, which is a very general n-dimensional strategy; the idea of virtual landmarks, which may be situated anywhere relative to, not necessarily on, the shape boundary; a new compactness measure that considers both the number of landmarks and the number of modes selected as independent variables. The first of three methods uses the c-scale shape descriptor, based on the new concept of curvature-scale, to automatically locate mathematical landmarks on the mean of the training shapes. The landmarks are propagated to the training shapes to establish correspondence among shapes. Since all shapes of the same family do not necessarily present exactly the same shape features, another novel method was devised that takes into account the real shape variability existing in the training set and that is guided by the strategy of equalization of the variance observed in the training set for selecting landmarks. By incorporating the above basic concepts into modelling, a third family of methods with numerous possibilities was developed, taking into account shape features, and the variability among shapes, while being easily generalized to the 3D space. Its output is multi-resolutional allowing landmark selection at any lower resolution trivially as a subset of those found at a higher resolution. The best strategy to use within the family will have to be determined according to the clinical application at hand. All methods were evaluated in terms of compactness on two data sets - 40 CT images of the liver and 40 MR images of the talus bone of the foot. Further, numerous artificial shapes with known salient points were also used for testing the accuracy of the proposed methods. The results show that, for the same number of landmarks, the proposed methods are more compact than manual and equally spaced annotations. Besides, the accuracy (in terms of false positives and negatives and the location of landmarks) of the proposed shape descriptor on artificial shapes is considerably superior to a state-of-the-art scale space approach to finding salient points on shapes

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more