Parseval networks improve deep nets' robustness to adversarial examples.
problem Improving deep neural networks' robustness to adversarial attacks.
method Constraining the Lipschitz constant and maintaining Parseval tight frames in weight matrices.
result Parseval networks maintain accuracy and robustness to adversarial examples compared to vanilla networks.
The paper explores Parseval frames on vector bundles, proving their existence for certain cases.
problem Existence of Parseval frames on vector bundles.
method Using G-bundles and algebraic topology, the authors prove the existence of Parseval frames for orientable vector bundles and provide conditions for smaller size frames. result The existence of Parseval frames for orientable vector bundles and conditions for smaller size frames.
Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…
Designs CNNs for better image reconstruction.
problem Image reconstruction from limited data.
method Parseval convolution operators and chaining of elementary modules.
result CNN-based algorithm yields better results than sparsity-based methods.
New method interprets ranked data on permutahedron graph.
problem Interpreting and exploiting structure in ranked data sets.
method Combining combinatorial representation theory and signal processing on graphs.
result Developed scalable transform method using Parseval frames.
Optimal square matrices for image approximation under translation and rotation.
problem Approximating images with translation and rotation invariant subspaces.
method Abstract harmonic analysis for constructing optimal square matrices.
result Optimal approximation of images with minimal quadratic error.
Simply connected spaces of tight frames identified.
problem Understanding the connectivity of spaces of tight frames.
method Viewing tight frames as elements of Stiefel manifolds and identifying simply connected spaces.
result Spaces of tight frames, including finite unit-norm tight frames, are simply connected.
Gradient descent constructs tight fusion frames.
problem Constructing tight fusion frames from prescribed subspaces.
method Gradient descent and symplectic geometry.
result Gradient descent can be used to construct tight fusion frames.
Frames for Rn can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word "frame" has a different meaning in the context of differential geometry and topology. A moving frame for the tang…
This paper interprets Dropout as creating an equiangular tight frame in autoencoders.
problem Understanding the success of Dropout in deep learning.
method Connecting Dropout to analog channel coding and equiangular tight frames (ETF).
result Optimizing autoencoders with dropout leads to an equiangular tight frame structure.
New method denoises graph signals using wavelets, scalable for large graphs.
problem Denoising graph signals with overcomplete tight frames and correlated noise.
method Data-driven wavelet tight frame, Stein's unbiased risk estimate, Chebyshev-Jackson polynomial approximations, Monte-Carlo strategy.
result Method scales to large graphs and finds applications in differential privacy.
Quantizes neural networks using frame theory for improved accuracy.
problem Improving neural network efficiency and accuracy through quantization.
method Sigma-Delta (ΣΔ) quantization with finite unit-norm tight frames. result Error bound between original and quantized neural networks derived.
Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.
problem Understanding isolation properties of geodesic planes in hyperbolic 3-manifolds.
method Quantitative estimates of geodesic planes in frame bundles, using tight areas and densities.
result Polynomial estimates of isolation properties with degree given by modified critical exponents.
We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure iff the corresponding group of deck transformations on S3 preserves a standard contact structure pointwise. We also relate univerally tight contact structures on 3-manifolds covered by S3 to the exceptional isomorph…
New wavelet frames constructed from reproducing kernels for continuous and discrete domains.
problem Generating wavelet frames on non-Euclidean structures.
method Spectral filtering of integral operators associated with reproducing kernels.
result Discrete frames as Monte Carlo estimates of continuous frames, with finite-sample rates derived.
The paper classifies tight contact structures on Seifert fiber spaces.
problem Classifying tight contact structures on Seifert fiber spaces.
method Using Legendrian surgery and convex surface theory.
result Tight contact structures on certain Seifert fiber spaces are classified.
It is well-known that a knot in a contact manifold (M,C) transverse to a trivialized contact structure possesses the natural framing given by the first of the trivialization vectors along the knot. If the Euler class eC∈H2(M) of C is nonzero, then C is nontrvivializable and the natural framing of transvers…
New framework aligns latent representations over-the-air using intelligent metasurfaces.
problem Heterogeneous transmitter-receiver models produce misaligned latent representations in semantic communication.
method Intelligent metasurfaces (SIM) emulate supervised and zero-shot semantic aligners directly in the wave domain.
result SIMs achieve up to 90% task accuracy in high SNR regimes, robust to low SNR.
FPCA optimizes fairness in target vectors' span.
problem Fairness in principal component analysis for multiple target vectors.
method Non-concave maximization of worst projected target norm using sub-gradient descent.
result Optimization landscape is benign with globally optimal local minima.
This is a second paper in a series devoted to the minimal unitary representation of O(p,q). By explicit methods from conformal geometry of pseudo-Riemannian manifolds, we find the branching law corresponding to restricting the minimal unitary representation to natural symmetric subgroups. In the case of purely discrete…
New deep learning methods improve CT image quality from few projections.
problem Sparse-view CT images suffer from streaking artifacts due to limited projections.
method Inspired by deep convolutional framelets, propose new U-Net variants that satisfy the frame condition.
result New U-Net variants provide better reconstruction performance for sparse-view CT.
The note evaluates different methods for option pricing using Shannon Wavelets.
problem Efficient computation of Shannon Wavelet coefficients for option pricing.
method Evaluation of cosine expansion, direct algorithms, and Filon quadrature.
result Filon quadrature is more efficient for computing Shannon Wavelet coefficients.
Paper tackles image reconstruction from limited data using polyhedral norms and convex regularizers.
problem Learning convex regularizers for image reconstruction from limited data.
method Imposes amplitude-equivariance, approximates functionals with polyhedral norms, identifies synthesis and analysis forms, proposes a trainable tight frame architecture.
result Proposed framework outperforms sparsity-based methods in denoising and biomedical image reconstruction.
Round surgery diagrams represent 3-manifolds in S3.
problem Representing and manipulating 3-manifolds in S3. method Introducing round surgery diagrams and defining moves to establish Kirby Calculus.
result Any 3-manifold can be obtained by a round surgery on a framed link in S3. We construct a simple topological invariant of certain 3-manifolds, including quotients of the 3-sphere by finite groups, based on the fact that the tangent bundle of an orientable 3-manifold is trivialisable. This invariant is strong enough to yield the classification of lens spaces of odd, prime order. We also use pr…
A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching 2-handles to convex boundaries of…
SSTQ improves privacy-preserving vector quantization with low communication cost.
problem Achieving local differential privacy in distributed optimization with low communication cost.
method Combines overcomplete equal-norm tight frames, coordinate subsampling, and privacy-aware one-dimensional quantization.
result Achieves optimal mean squared error scaling with only ⌈log2N⌉+b bits per client. A theory of feature geometry using spectral analysis of weight matrices.
problem Current methods decompose neural network activations into sparse linear features, losing geometric structure.
method Develops a theory by analyzing the spectra of weight-derived matrices, introducing the frame operator.
result Features collapse onto single eigenspaces, organizing into tight frames, and admit discrete classification.
The paper uncovers symmetries in large language models through layer-peeled optimization.
problem Understanding geometric structure in large language model weights and context embeddings.
method Constrained layer-peeled optimization program to analyze symmetries in next-token distributions.
result Symmetries in target next-token distributions are transferred to optimal model weights and context embeddings.
In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section…
New method for efficient proximal mapping of 1-path-norm in shallow networks.
problem Efficiently handling the 1-path-norm of shallow neural networks.
method Closed-form proximal operator for efficient computation and upper bound on Lipschitz constant.
result Proximal mapping allows robust training against adversarial perturbations.
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…
New geodesics found that are not tight but still have useful properties.
problem Understanding geodesics in curve complexes and Teichmüller spaces.
method Introducing and studying weak tight geodesics with canonical constructions.
result Found examples of weak tight geodesics with gaps between them.
We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…
Study tight contact structures on figure-eight knot surgeries.
problem Classify tight contact structures on surgeries of figure-eight knot.
method Analyzes surgeries on figure-eight knot, determining tightness, symplectic fillability, and universality.
result First classification of tight contact structures on surgeries of figure-eight knot.
The study identifies conditions for algorithms to have tight generalization bounds.
problem Understanding which algorithms have tight generalization bounds.
method Analyzing conditions that preclude tight generalization bounds and identifying stable algorithms.
result Stable algorithms have tight generalization bounds, while unstable ones do not.
Surgery on knots always admits a tight contact structure.
problem Understanding tight contact structures on knots after surgery.
method Smooth (-r)-surgery on knots, using Heegaard Floer contact invariant.
result Tight contact structures detected for all knots after surgery.
This work uses PAC-Bayes for structured prediction with ILE, yielding insights and algorithms.
problem Structured prediction with interdependent outputs and implicit loss embeddings.
method PAC-Bayes perspective applied to ILE framework, deriving generalization bounds and learning algorithms.
result Two learning algorithms derived from PAC-Bayes bounds, analyzed and implemented.
Tight maps was introduced along tight homomorphisms by Burger, Iozzi and Wienhard with aims towards maximal representations. In this paper we classify tight maps into classical Hermitian symmetric spaces and give a partial result for the exceptional spaces.
3-manifold triangulations are Golod and tight, proven through a topological characterization.
problem Understanding Golodness and tightness in 3-manifold triangulations.
method Topological characterization of a polyhedral product for a tight-neighborly manifold triangulation.
result Golodness and tightness are equivalent for 3-manifold triangulations.
The study finds many tight contact structures on hyperbolic 3-spheres.
problem Finding tight contact structures on hyperbolic 3-spheres.
method Constructing hyperbolic homology 3-spheres and analyzing their tight contact structures.
result Produces hyperbolic homology 3-spheres with multiple distinct tight contact structures.
Study tight contact structures on specific 3-manifolds.
problem Counting and constructing tight contact structures on plumbed 3-manifolds.
method Algorithm to construct stein diagrams for tight structures without Giroux torsion.
result Explicit algorithm to construct tight contact structures.
Classifies tight contact structures on surgeries of the Whitehead link.
problem Classifying tight contact structures on surgeries of the Whitehead link.
method Analyzes various surgeries on the Whitehead link to classify tight contact structures.
result Determines tight contact structures, Stein fillability, and virtually overtwisted properties.
5D contact structures are universally tight via Bourgeois construction.
problem Understanding tightness of 5D contact structures.
method Explicit construction of contact structures on VimesT2. result All constructed contact structures are universally tight.
New MCMC method tackles heavy tailed distributions using Fourier transforms.
problem Challenges in sampling from heavy tailed distributions using MCMC.
method Proposes Fourier transform MCMC to sample from densities with known Fourier transforms.
result Shows geometric ergodicity of the resulting Markov chain for heavy tailed distributions.
Study finds tight contact structures on many hyperbolic 3-manifolds.
problem Existence of tight contact structures on hyperbolic 3-manifolds.
method Dehn surgeries on hyperbolic surface bundles.
result Existence of infinitely many hyperbolic 3-manifolds with tight contact structures.
In \cite{confol} Y. Eliashberg and W. Thurston gave a definition of tight confoliations. We give an example of a tight confoliation ξ on T3 violating the Thurston-Bennequin inequalities. This answers a question from \cite{confol} negatively. Although the tightness of a confoliation does not imply the Thurston-Benn…
In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces. We also prove that any lens space admits only finitely many tight contact structures.