Shared classical randomness improves quantum generative models' output distributions.
problem Improving generative performance of shallow unitary quantum models.
method Introducing stochasticity into unitary quantum models via shared classical randomness.
result Shared classical randomness allows shallow unitary quantum models to represent a strictly larger family of distributions.
Full-capacity uRNNs improve performance over restricted-capacity ones.
problem Vanishing and exploding gradient issues in recurrent neural networks.
method Optimized full-capacity unitary recurrence matrices over all unitary matrices.
result Significantly improved performance compared to LSTMs and restricted-capacity uRNNs.
GORU combines unitary and gated RNNs for better long-term memory management.
problem Learning to effectively manage long-term memory in neural networks.
method Extending unitary RNNs with a gating mechanism to forget irrelevant information.
result GORU outperforms LSTMs, GRUs, and Unitary RNNs on long-term dependency tasks.
We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…
Contact group retracts to unitary subgroup.
problem Understanding contact structures on 3-sphere.
method Proving deformation retraction to unitary subgroup.
result Group of contactomorphisms retracts to U(2).
Researchers extend geometric quantization to complex Abelian Lie supergroups.
problem Quantization of super Kähler structures on complex Abelian Lie supergroups.
method Extended geometric quantization scheme to super Kähler setting, constructed unitary representation.
result Irreducible subrepresentations of the constructed representation are determined by the moment map.
Zeta functions for non-unitary twists are shown to have analytic continuation.
problem Analytic continuation of zeta functions for non-unitary twists.
method Analytic continuation for compact locally-symmetric spaces with non-unitary twists.
result Zeta functions admit analytic continuation as meromorphic functions.
Revises super unitary representations by relaxing Hilbert space definition.
problem Left-regular representations of super Lie groups are not super unitary.
method Weaken super Hilbert space definition and introduce a new metric.
result Left-regular representations of all super Lie groups are super unitary.
Study on determinants of unitary Brownian motion and their asymptotic laws.
problem Understanding determinants of unitary Brownian motion and their behavior over time.
method Using Stiefel fibration and skew-product decomposition of the Stiefel Brownian motion.
result Prove asymptotic laws for determinants of block entries of unitary Brownian motion.
We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial gen…
Study on learning quantum dynamics without direct interaction.
problem Learning quantum dynamics incoherently without direct interaction.
method Analyze sample complexity and prove bounds for incoherent learning.
result Prove that arbitrary measurements allow efficient learning of unitary processes incoherently.
New structure on unitary group of Hilbert space.
problem No specific problem stated; constructing a new structure.
method Constructing a Banach Poisson-Lie group structure.
result Banach Poisson-Lie group structure on unitary group of Hilbert space.
Study circle actions on unitary manifolds with discrete fixed points.
problem Understanding circle actions on compact unitary manifolds with discrete fixed points.
method Prove relationships between weights at fixed points and derive results regarding the first equivariant Chern class and Hirzebruch χ y χ_y χ y -genus. result Derive a multigraph encoding fixed point data, leading to new insights into unitary S 1 S^1 S 1 -manifolds. A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie al…
The study shows ergodicity of unitary frame flows on Kähler manifolds with specific curvature conditions.
problem Ergodicity of unitary frame flows on Kähler manifolds with negative holomorphic sectional curvature.
method Analysis of the unitary frame flow on the principal U(m)-bundle of unitary frames.
result For even-dimensional Kähler manifolds with negative λ(m)-pinched holomorphic sectional curvature, the unitary frame flow is ergodic and mixing.
UGConvs improve CNN accuracy with unitary transforms.
problem Improving CNN accuracy with richer representations.
method UGConvs combine group convolutions with unitary transforms.
result HadaNets achieve similar accuracy to circulant networks with lower complexity.
URNNs are as expressive as general RNNs with ReLU activations.
problem Expressiveness of URNNs compared to general RNNs.
method Input-output equivalence between URNNs and contractive RNNs with ReLU activations.
result URNNs are as expressive as general RNNs with ReLU activations.
Study asymptotics of unitary matrix elements in quantum mechanics.
problem Asymptotic behavior of unitary matrix elements in quantum mechanics.
method Uses Berezin-Toeplitz quantization and symplectic geometry.
result Recover asymptotics of Wigner's d-matrix elements for spin representations.
Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
New unitary representations for mapping class groups without almost invariant vectors.
problem Understanding unitary representations of mapping class groups.
method Space of measured foliations and Teichmüller space.
result None of the representations has almost invariant vectors.
Convexity proven for sums of angles of unitary paths.
problem Proving convexity of sums of eigenvalues of unitary matrices.
method Analyzing paths of unitary matrices and their angles, using operator norms.
result Sum of first m angles of unitary path is convex.
EUNNs improve RNN performance and efficiency.
problem Gradient explosion/vanishing and long-term correlations in RNNs.
method Unitary matrices with tunable representation capacity and O ( 1 ) \mathcal{O}(1) O ( 1 ) computational complexity. result EUNNs significantly outperform other RNNs and LSTMs in performance and training speed.
New unitary RNN architecture using complex Cayley transform outperforms existing methods.
problem Vanishing or exploding gradient problem in RNNs.
method Developed a unitary RNN architecture based on a complex scaled Cayley transform.
result scuRNN achieves comparable or better results than existing unitary RNNs.
Compactifies Minkowski space using unitary matrices.
problem Compactifying Minkowski space for quantum field theories.
method Using Cayley transform and unitary group $\U(2)$ .
result Defines interesting backgrounds for quantum field theories.
Study circumcenters in Finsler unitary groups with optimal convexity bounds.
problem Existence and convexity of circumcenters in Finsler unitary groups.
method Analysis of distance functions and p-Schatten norm on Lie algebra.
result Existence of circumcenters for sets with radius < π/2 in several metrics.
This paper introduces a submanifold of the moduli space of unitary representations of the fundamental group of a punctured sphere with fixed local monodromy. The submanifold is defined via products of involutions through Lagrangian subspaces. We show that the moduli space of Lagrangian representations is a Lagrangian s…
Defines unitary setting for quantum mechanics, explaining time evolution.
problem Completing quantum theory by defining unitary time evolution.
method Introduces geometric space with north and south poles, defines unitary time evolution as vector field flow.
result Unitary time evolution is explained as Lie group-Lie group algebra correspondence.
Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
Affirmatively answers Kosniowski conjecture for unitary S^1-manifolds.
problem Proving Kosniowski conjecture for specific types of manifolds.
method Analyzing unitary S^1-manifolds with isolated fixed points.
result 4χ(M) > dim(M) for the given manifolds, affirming the conjecture.
Let U c ( H ) = u : u i s u n i t a r y a n d u − 1 i s c o m p a c t U_c(H)={u: u is unitary and u-1 is compact} U c ( H ) = u : u i s u ni t a r y an d u − 1 i sco m p a c t stand for the unitary Fredholm group. We prove the following convexity result. Denote by d ∞ d_\infty d ∞ the rectifiable distance induced by the Finsler metric given by the operator norm in U c ( H ) U_c(H) U c ( H ) . If u 0 , u 1 , u ∈ U c ( H ) u_0,u_1,u\in U_c(H) u 0 , u 1 , u ∈ U c ( H ) and the geodesic β β β joining u 0 u_0 u 0 and u 1 u_1 u 1 in $U…
Quantum models learn unitary actions on entangled states from product states.
problem Generalization to out-of-distribution data in quantum machine learning.
method Proved out-of-distribution generalization for learning unitary actions.
result Learned unitary actions on entangled states from product states.
Geometric proof of contractibility of unitary group in strong topology.
problem Contractibility of unitary group in strong operator topology.
method Direct geometric proof and construction of special subspaces and operators.
result Direct geometric proof of contractibility theorem.
Generalizes Quillen's theorem to equivariant settings.
problem Classifying commutative formal group laws in equivariant bordism.
method Combines computations and detailed investigations of equivariant Lazard rings.
result Identifies the Z / 2 \mathbb{Z}/2 Z /2 -equivariant unitary bordism ring with the Z / 2 \mathbb{Z}/2 Z /2 -equivariant Lazard ring. Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
The paper confirms a conjecture about compact unitary manifolds with non-empty fixed points.
problem Bounding the dimension of compact unitary manifolds with non-empty fixed points.
method The approach involves confirming Kosniowski's conjecture for almost complex manifolds under specific weight conditions.
result The conjecture is confirmed for manifolds with specific types of weights.
Paper detects adversarial attacks in sound classification models.
problem Adversarial attacks threaten data-driven models, especially in sound classification.
method Detects adversarial subspaces in unitary vector domain using chordal distance and generalized Schur decomposition.
result Regularized logistic regression detector outperforms other approaches on benchmark datasets.
We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$ , together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce …
New biharmonic functions created on Lie groups.
problem Constructing explicit biharmonic functions on Lie groups.
method Developed a new scheme for constructing complex-valued biharmonic functions on Riemannian Lie groups.
result Manufactured infinite series of new solutions on S U ( n ) SU(n) S U ( n ) and showed applicability to S O ( n ) SO(n) S O ( n ) and S p ( n ) Sp(n) S p ( n ) . Paper presents a new VMBQC model with fewer parameters for better generative modeling.
problem Limited generative power of VMBQC due to more parameters than unitary models.
method Introduces a restricted VMBQC model with a single additional trainable parameter.
result Minimal extension of VMBQC model generates distributions not learnable by unitary models.
Paper uses Turaev-Viro TQFT to estimate 3-manifold genus.
problem Estimating the Heegaard genus of 3-manifolds.
method Turaev-Viro state sum TQFT and unitary modular category.
result Provides a lower bound for Heegaard genus using TQFT.
The large-N limit of Segal-Bargmann transform on spheres is studied.
problem Understanding the behavior of Segal-Bargmann transform on spheres as dimension increases.
method Analyzing the large-N limit of the transform on S N − 1 ( N ) S^{N-1}(\sqrt N) S N − 1 ( N ) , describing geometric models, and showing the transform remains unitary. result The limiting transform is still a unitary map from the limiting domain onto the limiting range.
Modeling functional data, this study uncovers the size-and-shape of functions under noisy observations.
problem Uncertainty in recovering a fixed effect function from noisy observations.
method Bayesian functional mixed model with priors on unitary transformations.
result It is possible to recover the size-and-shape of a square-integrable function μ μ μ . Paper improves tensor completion using unitary transforms.
problem Robust tensor completion for various datasets.
method Transformed tensor SVD with unitary matrices.
result Recovered images have better PSNR than traditional methods.
The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the…
Topological quantum computation with Fibonacci anyons relies on the possibility of efficiently generating unitary transformations upon pseudoparticles braiding. The crucial fact that such set of braids has a dense image in the unitary operations space is well known; in addition, the Solovay-Kitaev algorithm allows to a…
We describe the unitary globalization of cohomologically induced modules $A_{\fq}(λ)$ . The purpose of the paper is to give a geometric realization of the unitarizable modules. Our results do not constitute a proof of unitarity.
Classifies matrices in the quaternionic hyperbolic unitary group.
problem Understanding the structure of matrices in the quaternionic hyperbolic unitary group.
method Used complex representation and characteristic polynomial to study matrices.
result Computed the characteristic polynomial and studied its sign.
Study of unitary and groupoid orbits of normal operators, focusing on manifold structures and spectral conditions.
problem Understanding the manifold structures of orbits of normal operators under different norm topologies.
method Unified treatment of unitary and groupoid orbits, using moment maps and conditional expectations.
result Differentiable structures for orbits and necessary spectral conditions for norm closure and submanifold properties.