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.
We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all…
Private algorithms approximate matrices with private data.
problem Approximate matrices with same spectrum using private data.
method Differential privacy algorithms for unitary orbit optimization.
result Upper and lower bounds on approximation error.
Quantizes Toda systems using geometric methods.
problem Quantizing Toda systems with geometric quantization.
method Geometric quantization of Toda systems as a coadjoint orbit of a group of matrices.
result Found unitary and non-unitary finite dimensional quantum Hilbert spaces.
A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negative-definite Hermitian.…
Complex tensor factorization improves knowledge graph completion.
problem Automatically understanding and predicting missing relationships in large knowledge graphs.
method Use of complex-valued embeddings and unitary diagonalization.
result Complex embeddings lead to scalable and expressive models that outperform existing methods.
scoRNN improves RNN performance with simpler orthogonal weight matrices.
problem Vanishing and exploding gradients in RNNs.
method Parametrizing orthogonal recurrent weight matrices with a scaled Cayley transform.
result scoRNN achieves superior results with fewer parameters than other unitary RNNs.
We explore the class of triples (M, nabla, P) where M is a manifold, nabla is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G subgroup of GL(Rn) …
We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain…
Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.
problem Understanding properties of convex hulls of coadjoint orbits of compact groups.
method Introduce partial convex hulls and use them to define numerical invariants.
result Orbits with new invariants form rational convex polyhedral cones related to Littlewood-Richardson cones.
High-fidelity quantum simulations demonstrated on short-coherence hardware.
problem Short coherence times limit the depth of quantum algorithms.
method Fixed State Variational Fast Forwarding (fsVFF) algorithm.
result Simulations of 600 time steps possible, 150x longer than previous methods.
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).
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.
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…
Combines techniques to remove tameness condition in Morse-Smale flows.
problem Tameness condition in Morse-Smale flows.
method Combines Shilnikov's ODE techniques with Latschev's ideas.
result Removes tameness hypothesis from homotopy formula.
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 χy-genus. result Derive a multigraph encoding fixed point data, leading to new insights into unitary S1-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.
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.
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.
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) computational complexity. result EUNNs significantly outperform other RNNs and LSTMs in performance and training speed.
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.
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 Uc(H)=u:uisunitaryandu−1iscompact stand for the unitary Fredholm group. We prove the following convexity result. Denote by d∞ the rectifiable distance induced by the Finsler metric given by the operator norm in Uc(H). If u0,u1,u∈Uc(H) and the geodesic β joining u0 and u1 in $U…
This paper solves matrix blind joint block diagonalization with noise.
problem Identifying the diagonalizer and block diagonal structure of matrices under noise.
method Bi-block diagonalization method.
result The method can identify the exact solution under certain conditions.
Diagonalizes metrics of 3D Lorentzian manifolds.
problem Diagonalizing metrics of 3D Lorentzian manifolds.
method Applying the technique of moving frames.
result Every smooth Lorentzian 3-manifold admits an atlas with a diagonal metric.
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.
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-equivariant unitary bordism ring with the 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.
Study Ricci vector fields on 2D space with diagonal metrics.
problem Understanding Ricci vector fields on 2D space with specific metrics.
method Examined Ricci vector fields on R2 with a diagonal metric. result Characterized Ricci vector fields on R2 with a diagonal metric. Develops a novel stochastic algorithm for diagonal estimation of large matrices.
problem Efficient diagonal estimation for large or implicit matrices.
method Adaptive parameter selection in a stochastic algorithm.
result Lower bound on random query vectors needed for estimation.
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.
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.
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 …
Octagon map accelerates diagonal changes algorithm.
problem Improving the efficiency of diagonal changes algorithm.
method Octagon Farey map as an acceleration.
result Octagon map accelerates diagonal changes algorithm.