Abstract: Necessary and sufficient conditions for gradient flows of relative entropy in Lindblad equations.
problem Conditions for gradient flows in finite-dimensional Lindblad equations.
method Analyzes conditions for a finite-dimensional Lindblad equation to have a gradient flow structure for the von Neumann relative entropy.
result A finite-dimensional Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the BKM-detailed balance condition holds.
The von Neumann graph entropy (VNGE) facilitates measurement of information divergence and distance between graphs in a graph sequence. It has been successfully applied to various learning tasks driven by network-based data. While effective, VNGE is computationally demanding as it requires the full eigenspectrum of the…
New algorithms for clustering and dimension reduction using relative von Neumann entropy.
problem Clustering and dimension reduction for complex data sets.
method Construct graphs from data points, select graph maximizing relative von Neumann entropy, use eigenvectors for dimension reduction.
result Outperforms existing methods on non-trivial data sets.
On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q spaces for 1/p+1/q=1, we show that we can introduce the α-divergence, for αin (…
Study relative commutants in von Neumann algebras using contraction notions.
problem Understanding relative commutants in group and tracial crossed product von Neumann algebras.
method Introducing contraction notions to study relative commutants.
result Results applied to negatively curved groups and SL(d, Z).
It is shown that the Novikov inequalities for critical points of closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a corollary we obtain a vanishing theorem for L2 cohomology, generalizing a theorem of W. Lueck. We also prove that von Neumann Betti numbers coincide with the Nov…
Paper generalizes kernel mean embedding to von Neumann-algebra-valued measures.
problem Analyzing complex multivariate distributions and quantum mechanics.
method Generalizes kernel mean embedding to von Neumann-algebra-valued measures in reproducing kernel Hilbert modules.
result Injectivity and universality of the generalized KME are confirmed.
In this paper, we develop the notion of entropy for uniform hypergraphs via tensor theory. We employ the probability distribution of the generalized singular values, calculated from the higher-order singular value decomposition of the Laplacian tensors, to fit into the Shannon entropy formula. We show that this tensor …
A quantum state generation method that respects physical constraints.
problem Generating quantum states with complex-valued Hermitian, positive semi-definite, and trace one properties.
method Mirror diffusion model with von Neumann entropy to enforce structural constraints.
result Demonstrated effective generation of quantum states with conditional guidance.
We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's…
Making use of its smooth structure only, out of a connected oriented smooth 4-manifold a von Neumann algebra is constructed. It is geometric in the sense that is generated by local operators and as a special four dimensional phenomenon it contains all algebraic (i.e., formal or coming from a metric) curvature tensors…
We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.
Constructs a representation of the string 2-group on a von Neumann algebra.
problem Establishing a categorified spinor representation of the string 2-group.
method Using the Morita bicategory of von Neumann algebras, specifically the hyperfinite type III_1 factor.
result Demonstrates a categorification of the spinor representation.
In this paper we suggest a new general formalism for studying the invariants of polyhedra and manifolds comming from the theory of von Neumann algebras. First, we examine generality in which one may apply the construction of the extended abelian category, which was suggested in the previous publications of the author, …
We study the learnability of a class of compact operators known as Schatten--von Neumann operators. These operators between infinite-dimensional function spaces play a central role in a variety of applications in learning theory and inverse problems. We address the question of sample complexity of learning Schatten-von…
We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann rho-invariants associated with certain metabelian representations then so do both knots. As an application, we give a new example of an infinite family of knots which are linearly independent in the knot concorda…
Torsion objects of von Neumann categories describe the phenomen "spectrum near zero" discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such form can be represented as a discriminant form of a degenerate Hermitian form…
Gradient Descent Ascent converges to von-Neumann solution in hidden zero-sum games.
problem Understanding dynamics of zero-sum games with hidden structure.
method Gradient Descent Ascent applied to hidden zero-sum games with specific convex-concave structure.
result Gradient Descent Ascent converges to von-Neumann solution in strictly convex-concave hidden games.
Cone structures in quantum field theory linked to information geometry.
problem Understanding geometric structures in quantum field theory.
method Analyzing invariant cones under modular automorphism groups and their relation to Wishart laws.
result Explicit connection between CAH cones and Wishart laws.
The paper extends von Neumann's theory to normed modules and shows how they can be represented.
problem Understanding the structure of normed modules and their representability.
method Combining von Neumann's theory of liftings with Gigli's differential structure.
result Every separable normed module can be represented as sections of a measurable Banach bundle.
Entropy analysis via kernel methods for probabilistic inference.
problem Entropy analysis of probability distributions.
method Kernel methods and reproducing kernel Hilbert spaces for entropy estimation.
result New upper-bounds on log partition functions for probabilistic inference.
We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher-order von Neumann ρ-invariants by Cochran, Orr, and Teichner. We show that for a nonnegative integer n, if the connected sum of two n-solvable knots with coprime Alexander polynomials is slice, then each …
We provide criteria for self-adjointness and τ-Fredhomness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace τ. We extend the Callias-type index to operators acting on sections of…
We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in [0,∞] which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This all…
The paper defines a new Lie groupoid concept for infinite dimensions.
problem Obstacles in infinite-dimensional Lie groupoids.
method Introduces a new notion of 'bi-algebroid' for infinite dimensions.
result Recover partial Poisson manifolds and Banach Poisson Lie groups.
We explain new developments in classical knot theory in 3 and 4~dimensions, i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In dimension~3 we give a geometric interpretation of the Kontsevich integral (joint with Jim Conant), and in dimension 4 we introduce new concordance invariants using v…
A game-theoretic approach to multi-criteria ranking from ordinal data.
problem Ranking objects from ordinal data with multiple criteria.
method Generalizing von Neumann winner to multi-criteria setting using Blackwell's approachability.
result The Blackwell winner can be computed as a convex optimization problem and achieves near-optimal sample complexity.
We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold whe…
Suppose M is a compact manifold with boundary. Let N be a normal covering of M. Suppose (A,T) is an elliptic differential boundary value problem on M with lift (\tilde A,\tilde T) to N. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: these numbers are fi…
The aim of this work is to extend the capital growth theory developed by Kelly, Breiman, Cover and others to asset market models with transaction costs. We define a natural generalization of the notion of a numeraire portfolio proposed by Long and show how such portfolios can be used for constructing growth-optimal inv…
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…
We show that the L^2-torsion and the von Neumann rho-invariant give rise to commensurability invariants of knots.
New framework shows C∗-simplicity for groups without certain subalgebras.
problem Characterizing C∗-simplicity of groups. method Introducing confined subalgebras and Uniformly Recurrent States.
result A countable discrete group is C∗-simple if it has no non-trivial amenable confined subalgebras. New insights into mirror symmetry via Monge-Ampère domains and pre-Frobenius manifolds.
problem Exploring mirror symmetry using Landau-Ginzburg models and probability densities.
method Investigating Landau-Ginzburg models through Koopman-von Neumann's construction, showing existence of Monge-Ampère domains, and proving mirror pairs via Berglund-Hubsch-Krawitz construction.
result Existence of Monge-Ampère domains and their connection to pre-Frobenius manifolds.
Spinor bundle constructed on loop space for string manifolds.
problem Constructing spinor bundles for string manifolds.
method Using ideas from Stolz and Teichner, a spinor bundle and fusion product are defined for a string manifold.
result Existence of spinor bundle with fusion product on a manifold X if and only if X admits a string structure.
This work improves transferability by considering conditional distributions in feature representations.
problem Improving transferability across multiple domains by considering conditional distributions.
method Introducing von Neumann conditional divergence to quantify the functional dependence between features and desired response.
result Favorable performance in terms of smaller generalization error and less catastrophic forgetting.
Unified geometric framework for quantum states using dual number algebras.
problem Representing quantum states in a geometrically unified way.
method Smooth embeddings into higher-order dual number algebras and algebraic flows.
result Established nilpotent dual algebras as a geometric landscape for quantum kinematics.
Consider a compact locally symmetric space M of rank r, with fundamental group Γ. The von Neumann algebra $\vn(Γ)$ is the convolution algebra of functions f∈ℓ2(Γ) which act by left convolution on ℓ2(Γ). Let Tr be a totally geodesic flat torus of dimension r in M and let $Γ_0\cong\bb Z^r$ be t…
The abstract proves a conjecture about geometric structures in Calabi-Yau orbifolds.
problem Proving a conjecture about geometric structures in Calabi-Yau orbifolds.
method Using the Koopman--von Neumann formulation of Landau--Ginzburg theory and a Lagrangian torus fibration.
result The base of the SYZ fibration is a Monge--Ampère domain (the open simplex) for all Berglund--Hübsch--Krawitz mirror pairs.
We calculate the optimal solutions of the fully heterogeneous Von Neumann expansion problem with N processes and P goods in the limit N→∞. This model provides an elementary description of the growth of a production economy in the long run. The system turns from a contracting to an expanding phase as N in…
In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L^2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy inva…
We discuss an infinite class of metabelian Von Neumann rho-invariants. Each one is a homomorphism from the monoid of knots to the real line. In general they are not well defined on the concordance group. Nonetheless, we show that they pass to well defined homomorphisms from the subgroup of the concordance group generat…
Constructs a fusion product on spinor bundle over loop space.
problem Describes a fusion product on the spinor bundle over loop space.
method Uses Connes fusion of von Neumann bimodules and string structures.
result Establishes a novel relation between string structures, loop fusion, and Connes fusion of Fock spaces.
We study a relation between the Hecke groups and the index of subfactors in a von Neumann algebra. Such a problem was raised by V. F. R. Jones. We solve the problem using the notion of a cluster C*-algebra.
We present a derivation of the Kullback Leibler (KL)-Divergence (also known as Relative Entropy) for the von Mises Fisher (VMF) Distribution in d-dimensions.
Study shows Fredholmness of a Lorentzian Dirac operator on spacetimes.
problem Analyzing Fredholmness of a Lorentzian Dirac operator on spacetimes.
method Investigates Fredholmness of a (spatial) Γ-invariant Lorentzian Dirac operator under (anti) Atiyah-Patodi-Singer boundary conditions.
result Demonstrates Fredholmness of the operator in the von Neumann sense.
In this paper, we discuss relations among several invariants of 3-manifolds including Meyer's function, the eta-invariant, the von Neumann rho-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.
Study of Gaussian distributions using entropic Gromov-Wasserstein and inner product Gromov-Wasserstein.
problem Optimal transportation between Gaussian distributions with different dimensions.
method Entropic Gromov-Wasserstein and inner product Gromov-Wasserstein, with closed-form expressions and von Neumann's trace inequality.
result Closed-form expressions for the entropic IGW and its unbalanced variant between Gaussian distributions.