We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.
Lusztig [Section 4] LusIntro used a flow similar to @math to show that @math is contractible. Our flow can be thought of as an affine (or loop group) analogue of his flow, and is closely related to the whirl matrices of @cite_29 . We also remark that Ayala, Kliemann, and San Martin @cite_6 used the language of control theory to give an alternative development in type @math of Lusztig's theory of total positivity. In that context, @math ( @math ) lies in the interior of the compression semigroup of @math , and @math is its attractor .
@cite_29 @cite_6
2963963365 2118972993 2542071751 2949560198
Abstract This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group G L n ( R [ t , t − 1 ] ) , and for the formal loop group G L n ( R ( ( t ) ) ) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeros of the Edrei–Thoma theorem classifying totally positive functions (corresponding to our case n = 1 ). We give a solution to the “factorization problem” using limits of ratios of minors. This is in a similar spirit to the Berenstein–Fomin–Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi–Yamada’s study of discrete Painleve dynamical systems and Berenstein–Kazhdan’s study of geometric crystals. Abstract Since the seminal work of [Sussman, M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 1994;114:146–59] on coupling the level set method of [Osher S, Sethian J. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 1988;79:12–49] to the equations for two-phase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 1994;114:146–59] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both of its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as Hamilton–Jacobi WENO [Jiang G-S, Peng D. Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J Sci Comput 2000;21:2126–43], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [Enright D, Fedkiw R, Ferziger J, Mitchell I. A hybrid particle level set method for improved interface capturing. J Comput Phys 2002;183:83–116] and the coupled level set volume of fluid method [Sussman M, Puckett EG. A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J Comput Phys 2000;162:301–37], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [Losasso F, Gibou F, Fedkiw R. Simulating water and smoke with an octree data structure, ACM Trans Graph (SIGGRAPH Proc) 2004;23:457–62]. Motivated by online recommendation and advertising systems, we consider a causal model for stochastic contextual bandits with a latent low-dimensional confounder. In our model, there are @math observed contexts and @math arms of the bandit. The observed context influences the reward obtained through a latent confounder variable with cardinality @math ( @math ). The arm choice and the latent confounder causally determines the reward while the observed context is correlated with the confounder. Under this model, the @math mean reward matrix @math (for each context in @math and each arm in @math ) factorizes into non-negative factors @math ( @math ) and @math ( @math ). This insight enables us to propose an @math -greedy NMF-Bandit algorithm that designs a sequence of interventions (selecting specific arms), that achieves a balance between learning this low-dimensional structure and selecting the best arm to minimize regret. Our algorithm achieves a regret of @math at time @math , as compared to @math for conventional contextual bandits, assuming a constant gap between the best arm and the rest for each context. These guarantees are obtained under mild sufficiency conditions on the factors that are weaker versions of the well-known Statistical RIP condition. We further propose a class of generative models that satisfy our sufficient conditions, and derive a lower bound of @math . These are the first regret guarantees for online matrix completion with bandit feedback, when the rank is greater than one. We further compare the performance of our algorithm with the state of the art, on synthetic and real world data-sets. We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting by-product of our analysis is to establish the existence of ldquoprobabilistic expansionrdquo in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting.