Optimality of Linear Sketching Under Modular Updates

We study the relation between streaming algorithms and linear sketching algorithms, in the context of binary updates. We show that for inputs in n dimensions, the existence of efficient streaming algorithms which can process Omega(n^2) updates implies efficient linear sketching algorithms with comparable cost. This improves upon the previous work of Li, Nguyen and Woodruff [Yi Li et al., 2014] and Ai, Hu, Li and Woodruff [Yuqing Ai et al., 2016] which required a triple-exponential number of updates to achieve a similar result for updates over integers. We extend our results to updates modulo p for integers p >= 2, and to approximation instead of exact computation

Approximate F_2-Sketching of Valuation Functions

We study the problem of constructing a linear sketch of minimum dimension that allows approximation of a given real-valued function f : F_2^n - > R with small expected squared error. We develop a general theory of linear sketching for such functions through which we analyze their dimension for most commonly studied types of valuation functions: additive, budget-additive, coverage, alpha-Lipschitz submodular and matroid rank functions. This gives a characterization of how many bits of information have to be stored about the input x so that one can compute f under additive updates to its coordinates. Our results are tight in most cases and we also give extensions to the distributional version of the problem where the input x in F_2^n is generated uniformly at random. Using known connections with dynamic streaming algorithms, both upper and lower bounds on dimension obtained in our work extend to the space complexity of algorithms evaluating f(x) under long sequences of additive updates to the input x presented as a stream. Similar results hold for simultaneous communication in a distributed setting

Non-Convex and Geometric Methods for Tomography and Label Learning

Data labeling is a fundamental problem of mathematical data analysis in which each data point is assigned exactly one single label (prototype) from a finite predefined set. In this thesis we study two challenging extensions, where either the input data cannot be observed directly or prototypes are not available beforehand. The main application of the first setting is discrete tomography. We propose several non-convex variational as well as smooth geometric approaches to joint image label assignment and reconstruction from indirect measurements with known prototypes. In particular, we consider spatial regularization of assignments, based on the KL-divergence, which takes into account the smooth geometry of discrete probability distributions endowed with the Fisher-Rao (information) metric, i.e. the assignment manifold. Finally, the geometric point of view leads to a smooth flow evolving on a Riemannian submanifold including the tomographic projection constraints directly into the geometry of assignments. Furthermore we investigate corresponding implicit numerical schemes which amount to solving a sequence of convex problems. Likewise, for the second setting, when the prototypes are absent, we introduce and study a smooth dynamical system for unsupervised data labeling which evolves by geometric integration on the assignment manifold. Rigorously abstracting from ``data-label'' to ``data-data'' decisions leads to interpretable low-rank data representations, which themselves are parameterized by label assignments. The resulting self-assignment flow simultaneously performs learning of latent prototypes in the very same framework while they are used for inference. Moreover, a single parameter, the scale of regularization in terms of spatial context, drives the entire process. By smooth geodesic interpolation between different normalizations of self-assignment matrices on the positive definite matrix manifold, a one-parameter family of self-assignment flows is defined. Accordingly, the proposed approach can be characterized from different viewpoints such as discrete optimal transport, normalized spectral cuts and combinatorial optimization by completely positive factorizations, each with additional built-in spatial regularization

Open Problems in (Hyper)Graph Decomposition

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition