Modified proof constructs holomorphic quilts on closed surfaces.
problem Compare Lagrangian Floer theory with quilted Lagrangian Floer theory.
method Modified proof of holomorphic quilts from Wehrheim and Woodward.
result Supports Bottman and Wehrheim's conjecture on isomorphism.
Proves uniqueness of holomorphic quilts on surfaces.
problem Computing boundary maps of immersed Lagrangian Floer chain groups.
method Constructs holomorphic quilts from bigons on surfaces.
result Uniqueness of holomorphic quilts provides a combinatorial method for computing boundary maps.
New mechanism improves privacy in time series data.
problem Lack of composition properties in inferential privacy.
method Study of Pufferfish mechanism for time series data.
result Markov Quilt Mechanism has strong composition properties.
Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.
John Conway created pairs of domains that sound the same for a special kind of music.
problem Creating domains that sound the same for a special kind of music.
method Using his theory of quilts, Conway developed pairs of glueing diagrams.
result Conway's pairs of domains are isospectral for the Laplace operator.
We define relative Floer theoretic invariants arising from 'quilted pseudo-holomorphic surfaces': Collections of pseudoholomorphic maps to various target spaces with 'seam conditions' in Lagrangian correspondences. As application we construct a morphism on quantum homology associated to any monotone Lagrangian correspo…
New methods estimate brain connectivity from calcium imaging data with missing data.
problem Estimating functional neuronal connectivity from calcium imaging data with missing data.
method Two approaches for nonparanormal Graph Quilting based on the Gaussian copula graphical model.
result Our methods yield more meaningful functional connectivity estimates than existing Gaussian graph quilting methods.
Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence …
In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretic…
Cluster Quilting clusters fragmented data sets for neuroscience and genomics.
problem Clustering fragmented data sets in neuroscience and genomics.
method Cluster Quilting method using patch ordering, patchwise SVD, sequential linear mapping, and k-means.
result Cluster Quilting discovers more accurate clusters than other methods.
Let G be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal G-bundle, P\to Σ, over a compact oriented surface, Σ, carries a Poisson structure. If we trivialize P over a finite number of points on the boundary of Σ, then the moduli space carries a q…
New method estimates neuronal connectivity from partially observed data.
problem Estimating neuronal connectivity from partially observed data.
method Two-step approach: low-rank covariance completion followed by graph structure estimation.
result Graph selection consistency demonstrated for one approach.
We describe the first sub-quadratic sampling algorithm for the Multiplicative Attribute Graph Model (MAGM) of Kim and Leskovec (2010). We exploit the close connection between MAGM and the Kronecker Product Graph Model (KPGM) of Leskovec et al. (2010), and show that to sample a graph from a MAGM it suffices to sample sm…
We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. We give applications to calculations of Floer cohomology, displaceability of Lagrangian correspond…
We realize Stasheff's multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.
We use the theory of pseudo-holomorphic quilts to establish a counterpart, in symplectic Floer homology, to the Gysin sequence for the homology of a sphere-bundle. In a motivating class of examples, this "symplectic Gysin sequence" is precisely analogous to an exact sequence describing the behaviour of Seiberg-Witten m…
We extend Perutz's Lagrangian matching invariants to 3-manifolds which are not necessarily fibred using the technology of holomorphic quilts. We prove an isomorphism of these invariants with Ozsvath-Szabo's Heegaard Floer invariants for certain extremal spin^c structures. As applications, we give new calculations of He…
Study on singularities of Lagrangian immersions with applications in Floer theory.
problem Understanding singularities of Lagrangian immersions.
method Applying Hamiltonian isotopy in the Weinstein tubular neighbourhood to express singular points as fold points with cusp points.
result Local expression of singular points of Lagrangian immersions as fold points with cusp points.
Develops a new method for equivariant Lagrangian Floer homology using symplectic homotopy quotients.
problem Constructing equivariant Lagrangian Floer homology for symplectic manifolds with group actions.
method Using symplectic homotopy quotients involving cotangent bundles of an approximation of EG, and Wehrheim and Woodward's theory of quilts. result Shows that the constructed groups are independent of auxiliary choices and are H∗(BG)-bimodules. This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S^2 union T^2, regarded as mo…
New knot homology theory from symplectic geometry.
problem Developing a symplectic counterpart to instanton knot homology.
method Using symplectic character varieties and Floer homology, a new invariant for knots in 3-manifolds is constructed.
result Symplectic instanton knot homology (SIK) is a new invariant of knots and links in 3-manifolds.
In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclas…
Study Markov cubature rules for polynomial processes.
problem Tractability of path-dependent tasks in polynomial process models.
method Discretizations using finite state Markov processes with moment matching conditions.
result Markov cubature rules aid American option pricing.
The paper estimates key metrics for linear models with Markov or hidden Markov sources.
problem Estimating free energy, mutual information, and MMSE for linear models with specific signal priors.
method Replica analysis in statistical physics, focusing on Markov and hidden Markov sources.
result The linear model with Markov or hidden Markov sources can be simplified into decoupled AWGN channels.
New method improves convergence of gradient descent for non-convex, non-reversible Markov chains.
problem Improving convergence of gradient descent for non-convex, non-reversible Markov chains.
method Introducing a new technique that varies the mixing levels of the Markov chains to establish non-ergodic convergence under wider step sizes.
result Established non-ergodic convergence for non-convex problems and non-reversible finite-state Markov chains.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
Expands Hidden Markov Model to include Markov chain observations.
problem Handling Markov chain observations in Hidden Markov Models.
method Developed Expectation-Maximization algorithm and Viterbi algorithm analogs.
result Estimates transition probabilities for hidden states and observations.
Study approximates financial market with discrete-time models.
problem Approximating continuous-time financial market models with discrete-time.
method Constructs discrete-time market models with Markov switching and proves convergence.
result Discrete-time models converge to continuous-time Black-Scholes model with Markov switching.
Paper tests Markov assumption in sequential decision making.
problem Testing the Markov assumption in sequential decision making.
method Forward-Backward Learning procedure to test MA without assuming parametric forms.
result The proposed test plays a crucial role in identifying optimal policies in complex decision processes.
New neural processes use stacked Markov operators to improve flexibility.
problem Improving flexibility in neural processes.
method Stacking neural parameterized Markov transition operators in function space.
result MNPs outperform baseline models on various tasks.
New algorithms for RL in Markov games with independent linear function approximation, breaking the curse of multiagents.
problem Tackles the challenge of learning Markov equilibria in large state space Markov games with multiple agents.
method Proposes independent linear Markov games and designs new algorithms for learning Markov coarse correlated equilibria and Markov correlated equilibria with polynomial sample complexity.
result Breaks the curse of multiagents by achieving sample complexity bounds that scale polynomially with each agent's function class complexity.
The paper bounds generalization errors for deep neural networks with Markov datasets.
problem Bounding generalization errors for deep learning with Markov datasets.
method Developed new symmetrization inequalities for Markov chains, using spectral gap of the infinitesimal generator.
result Derived upper bounds on generalization errors for deep neural networks with Markov datasets.
The paper extends Alexander and Markov theories to generalized knot theories.
problem Defining Alexander and Markov theories for generalized knot theories.
method Extending existing theories to new knot types.
result Alexander and Markov theories can be applied to generalized knot theories.
The paper constructs Markov partitions for geodesic flow on hyperbolic surfaces.
problem Understanding Markov partitions for general hyperbolic flows.
method Rigorous construction of Markov partitions for geodesic flow on Riemann surfaces of constant negative curvature.
result Explicit forms of rectangles and local cross sections provided for the geodesic flow.
Study nonparametric estimator for Markov chain transition matrices in offline setting.
problem Estimating transition matrices of finite controlled Markov chains from logged data.
method Developed sample complexity bounds and conditions for minimaxity.
result Achieving certain statistical risk requires balancing mixing properties and sample size.
We rephrase Gromov's definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum X, $\dim_{\Z_{(p)}}X=\dim_{\Q}X$ for all but finitely many primes p where Z(p) is the localization…
This paper introduces a new method for optimizing large-scale problems using Markov chain block updates.
problem Optimizing large-scale problems with efficient and natural block selection.
method Markov chain block coordinate descent (BCD) for optimization.
result The method converges for minimizing Lipschitz differentiable functions, with sublinear and linear convergence rates for convex and strongly convex functions, respectively.
New method estimates convergence bounds for nonlinear Markov chains.
problem Difficulty in describing properties of nonlinear Markov chains.
method Coupling Markov chains to reconstitute distribution relationships and estimate convergence bounds.
result Estimation of convergence bounds is more precise than existing results.
The paper provides concentration inequalities for Markov chain variance estimators.
problem Estimating the variance of Markov chains with concentration properties.
method Martingale decomposition method for uniformly geometrically ergodic Markov chains.
result Explicit control of the p-th moment of the OBM estimator difference and dependence on p and mixing time.
Proves Alexander- and Markov-type theorems for virtual trivalent braids.
problem Classifying virtual trivalent braids and graphs.
method Two versions of the Markov-type theorem: algebraic and L-move based.
result Established new theorems for virtual trivalent braids.
A new Markov theorem for 4D ribbon torus-links.
problem Describing isotopic links in R4. method Develops a theorem for ribbon torus-links in B3imesS1. result First step towards a 4D Markov theorem.
A new method simulates a lazy version of a Markov chain for empirical inference.
problem Estimating and testing unknown Markov chains with limited data.
method Simulates an α-lazy version of an unknown Markov chain, making it ergodic.
result The pseudo spectral gap can be applied to non-ergodic Markov chains.
Abstract reviews Markov processes with jumps on manifolds and Lie groups.
problem Analyzing Markov processes with jumps in geometric settings.
method Stochastic differential equations, Courrège theorem, invariant Markov processes.
result Developments in Lie groups and manifolds under various actions.
The paper calculates fair strike for variance swaps on time-changed Markov processes.
problem Calculating fair strike for variance swaps on time-changed Markov processes.
method Proving the fair strike equals the price of a European contract and solving the integro-differential equation.
result The fair strike for variance swaps can be computed explicitly for certain Markov processes.
This paper reviews recent advances in Bayesian nonparametric techniques for constructing and performing inference in infinite hidden Markov models. We focus on variants of Bayesian nonparametric hidden Markov models that enhance a posteriori state-persistence in particular. This paper also introduces a new Bayesian non…
Spectral methods reduce the complexity of Markov processes.
problem Modeling and simplifying state-transition systems.
method Spectral decomposition and state aggregation.
result Developed methods to estimate low-rank Markov models.
New method adds user constraints to Markov chains for better data reduction.
problem No systematic framework to impose user-defined constraints on Markov chains.
method Path entropy maximization to derive transition probabilities with user constraints.
result Improved nonlinear dimensionality reduction with user-prescribed constraints.
Improved pricing method for Asian options under Markov processes.
problem Pricing Asian options under Markov processes.
method Explicitly carried out inverse Z-transform and inverse Laplace transform for discretely and continuously monitored cases.
result Explicit single Laplace transforms improve efficiency.