Randomized Dynamic Mode Decomposition

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of `big data'. The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a fast decaying singular value spectrum

Fast and Simple Modular Subset Sum

We revisit the Subset Sum problem over the finite cyclic group $\mathbb{Z}_m$ for some given integer $m$. A series of recent works has provided asymptotically optimal algorithms for this problem under the Strong Exponential Time Hypothesis. Koiliaris and Xu (SODA'17, TALG'19) gave a deterministic algorithm running in time $\tilde{O}(m^{5/4})$, which was later improved to $O(m \log^7 m)$ randomized time by Axiotis et al. (SODA'19). In this work, we present two simple algorithms for the Modular Subset Sum problem running in near-linear time in $m$, both efficiently implementing Bellman's iteration over $\mathbb{Z}_m$. The first one is a randomized algorithm running in time $O(m\log^2 m)$, that is based solely on rolling hash and an elementary data-structure for prefix sums; to illustrate its simplicity we provide a short and efficient implementation of the algorithm in Python. Our second solution is a deterministic algorithm running in time $O(m\ \mathrm{polylog}\ m)$, that uses dynamic data structures for string manipulation. We further show that the techniques developed in this work can also lead to simple algorithms for the All Pairs Non-Decreasing Paths Problem (APNP) on undirected graphs, matching the asymptotically optimal running time of $\tilde{O}(n^2)$ provided in the recent work of Duan et al. (ICALP'19)

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

Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201

Faster Deterministic Modular Subset Sum

We consider the Modular Subset Sum problem: given a multiset X of integers from ?_m and a target integer t, decide if there exists a subset of X with a sum equal to t (mod m). Recent independent works by Cardinal and Iacono (SOSA\u2721), and Axiotis et al. (SOSA\u2721) provided simple and near-linear algorithms for this problem. Cardinal and Iacono gave a randomized algorithm that runs in ?(m log m) time, while Axiotis et al. gave a deterministic algorithm that runs in ?(m polylog m) time. Both results work by reduction to a text problem, which is solved using a dynamic strings data structure. In this work, we develop a simple data structure, designed specifically to handle the text problem that arises in the algorithms for Modular Subset Sum. Our data structure, which we call the shift-tree, is a simple variant of a segment tree. We provide both a hashing-based and a deterministic variant of the shift-trees. We then apply our data structure to the Modular Subset Sum problem and obtain two algorithms. The first algorithm is Monte-Carlo randomized and matches the ?(m log m) runtime of the Las-Vegas algorithm by Cardinal and Iacono. The second algorithm is fully deterministic and runs in ?(m log m ? ?(m)) time, where ? is the inverse Ackermann function