We construct 2-dimensional CAT(-1) groups which contain free subgroups with arbitrary iterated exponential distortion, and with distortion higher than any iterated exponential.
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Study stabilizers in handlebody group; meridians are undistorted, others are exponentially distorted.
problem Geometric properties of stabilizers in handlebody group.
method Analysis of stabilizers of meridians and primitive curves/annuli.
result Stabilizers of meridians are undistorted, others are exponentially distorted.
The study shows exponential distortion in virtually special groups containing free subgroups.
problem Understanding distortion in virtually special groups containing free subgroups.
method Constructing examples of virtually special groups with finite rank free subgroups.
result Distortion functions grow like exp^k(x^m) and can be superexponential.
The paper calculates subgroup distortions in 3-manifold groups.
problem Understanding subgroup distortions in 3-manifold groups.
method Computed all finitely generated subgroups of finitely generated 3-manifold groups and analyzed their distortions.
result Subgroup distortions in 3-manifold groups are linear, quadratic, exponential, or double exponential.
Let be a properly immersed --injective surface in a non-geometric --manifold . We compute the distortion of in and show that how it is related to separability of in . The only possibility of the distortion is linear, quadratic, exponential, an…
ReLU networks don't exponentially distort curve lengths as previously thought.
problem Understanding how neural networks distort curve lengths with depth.
method Analyzing expected length distortion of ReLU networks with random initialization.
result Expected length distortion does not grow with depth, and shrinks slightly.
Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.
Extends likelihood ratio exponential families to analyze various optimization methods.
problem Analyzing optimization methods like rate-distortion and information bottleneck.
method Linking geometric mixture paths to exponential families and using hypothesis testing.
result Provides a common mathematical framework for understanding these methods.
We prove that each Torelli group of an orientable surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus-Farb-Putman's techniques. Further we show that the distortion of each Torelli group in the level mapping class group is the same as that o…
We prove that various subgroups of the mapping class group of a surface are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute low…
We show that the mapping class group of a handlebody of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface. The same holds true for solid tori with at least two marked points or spots.
High-dimensional diffusion models suffer from distorted samples due to CFG.
problem Distortions in high-dimensional guided diffusion models.
method Analytical tools from statistical physics, dynamic mean-field theory.
result Distortions arise in high-dimensional settings due to class separability issues.
Circumventing the Curse of Dimensionality in Prediction: Causal Rate-Distortion for Infinite-Order Markov Processescond-mat.stat-mech
Predictive rate-distortion analysis suffers from the curse of dimensionality: clustering arbitrarily long pasts to retain information about arbitrarily long futures requires resources that typically grow exponentially with length. The challenge is compounded for infinite-order Markov processes, since conditioning on fi…
We propose a definition of quasi-local mass based on the Penrose Inequality. Two further definitions are given by measuring distortions of the exponential map.
Revises Gauss's Lemma using metrical distortion and differential slip.
problem Revising Gauss's Lemma in Riemannian geometry.
method Defining metrical distortion and differential slip, showing their geometric implications.
result Geodesically radial volume and length preservation properties.
Lossy Compression via Sparse Linear Regression: Computationally Efficient Encoding and Decodingcs.IT
We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. The codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix. The proposed encoding algorithm sequentially chooses columns of the design matrix to suc…
Robust Bregman Clusteringmath.ST
A robust clustering method for noisy data using Bregman divergences.
problem Clustering data corrupted with clutter noise.
method k-means type method based on Bregman divergences with a trimming approach.
result Empirically optimal codebook converges to an optimal codebook in the distortion sense.
Model resolves asset pricing puzzles with price-impact.
problem Asset pricing puzzles like interest rate, stock-price volatility, and equity premium.
method Closed-form equilibrium model with exponential investors trading continuously and experiencing price-impact.
result Price-impact amplifies risk-sharing distortions, resolving puzzles.
Linear progress in fibresmath.GT
Linear progress observed in fibered 3-manifold embeddings.
problem Understanding the geometric distortion of fibered embeddings.
method Verification of linear progress along typical rays using geometry and ergodic theory.
result A typical ray in the fiber makes linear progress in the ambient metric.
Boosted decision trees typically yield good accuracy, precision, and ROC area. However, because the outputs from boosting are not well calibrated posterior probabilities, boosting yields poor squared error and cross-entropy. We empirically demonstrate why AdaBoost predicts distorted probabilities and examine three cali…
There is a consensus that human and non-human subjects experience temporal distortions in many stages of their perceptual and decision-making systems. Similarly, intertemporal choice research has shown that decision-makers undervalue future outcomes relative to immediate ones. Here we combine techniques from informatio…
Sequence distortion measures large-scale distances in metric spaces.
problem Classifying and comparing large-scale distances in metric spaces.
method Introducing sequence distortion spectrum and defining -distorted sequences.
result Different rate functions yield distinct sequence distortions in various metric spaces.
Optimizes embedding accuracy for data variance and error.
problem Efficiently embedding data while minimizing distortion.
method Uses Johnson-Lindenstrauss embeddings with orthogonal matrices and singular-value latent variables.
result Achieves best accuracy in variance, mean-squared error, and length distortion.
Vertex distortion detects if a knot is unknot.
problem Determining if a knot is the unknot.
method Using Denne-Sullivan's bound on Gromov distortion, the vertex distortion of nontrivial lattice knots is bounded. Then, it is shown that trivial vertex distortion implies the unknot.
result The conjecture that trivial vertex distortion implies the unknot is proven.
Hyperbolic space outperforms Euclidean in learning hierarchical data.
problem Learning hierarchical data in Euclidean space requires exponentially many samples.
method Established geometric obstruction in Euclidean space and showed hyperbolic space's advantage.
result Hyperbolic space enables learning with samples, matching information-theoretic optimum.
New method corrects complex distortions in single view images.
problem Complex distortions in images, especially those caused by refractive surfaces.
method Differentiable image sampling and semantic information augmentation.
result Model can estimate and correct highly complex distortions.
Algorithm finds optimal affine transformation to minimize overall distortion.
problem Minimizing distortion in affine transformations.
method Riemannian geometry approach to define and minimize distortion.
result Mean distorting transformation found for minimizing overall distortion.
Noise-Calibrated Inference from Differentially Private Sufficient Statistics in Exponential Familiescs.LG
This paper provides a method for noise-calibrated inference from DP synthetic data.
problem Inference from DP synthetic data is often miscalibrated and lacks principled uncertainty quantification.
method Release DP sufficient statistics, perform noise-calibrated likelihood-based inference, and optional synthetic data generation.
result Asymptotic normality and valid confidence intervals for the plug-in DP MLE.
Vertex distortion measures how far lattice knots deviate from straight lines.
problem Measuring how much lattice knots deviate from straight paths.
method Analogous to smooth knots, study vertex distortion in lattice knots.
result Vertex distortion is 1 only for the unknot and can be arbitrarily high.
We consider the problem of distortion minimal morphing of -dimensional compact connected oriented smooth manifolds without boundary embedded in . Distortion involves bending and stretching. In this paper, minimal distortion (with respect to stretching) is defined as the infinitesimal relative change in vol…
This paper shows how to calculate risk measures for sums of two counter-monotonic risks.
problem Calculating risk measures for sums of two counter-monotonic risks.
method Using a fixed distortion function and expressing the risk measure of a sum as the sum of two related measures of the marginals.
result The risk measure of a sum of two counter-monotonic risks can be expressed as the sum of two related distortion risk measures of the marginals.
A new method compresses point clouds efficiently, outperforming existing techniques.
problem Efficiently compressing large point cloud datasets for VR applications.
method Learned convolutional transforms and uniform quantization for joint rate and distortion optimization.
result Significant rate-distortion improvement (51.5% BDBR savings) on Microsoft Voxelized Upper Bodies dataset.
A new family of stochastic dominance orders based on distortion functions.
problem Determining a continuum of dominance relations for risk assessment.
method Introducing H-distorted stochastic dominance, a generalized family of stochastic orders.
result Power-distorted stochastic dominance is particularly appealing due to its simplicity and statistical interpretations.
Study distortion risk measures for step-weighted distributions.
problem Analyzing risk measures for specific distribution types.
method Investigate distortion risk measures of step-weighted distributions.
result Developed methods for calculating risk measures.
The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed any closed curve has distortion at least pi/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5pi/3; examples show that distortion under 7.16 suffices to build a trefoil kno…
We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with c…
Given a simplicial complex , we consider several notions of geometric complexity of embeddings of in a Euclidean space : thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any -complex with simplices which topologically…
Computed distortion coefficients for the α-Grushin plane.
problem Analyzing the distortion coefficients of the α-Grushin plane.
method Using generalised trigonometric functions and synthetic curvature conditions.
result Estimates for distortion coefficients and a curvature condition conjecture.
Study on risk measures using distorted Choquet integrals with random distortions.
problem Developing risk measures under random distortions of capacities.
method Introducing and analyzing randomly distorted Choquet integrals with respect to a distorted capacity, establishing properties and providing representations.
result Representation of comonotonic additive conditional risk measures using G-randomly distorted Choquet integrals.
Sharp bounds on distortion of surfaces in 3D space.
problem Finding the minimum distortion of surfaces in 3D space.
method Analyzing convex embedded 2-spheres and surfaces of positive genus.
result π/2 is a sharp lower bound on the distortion of surfaces of positive genus.
We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, ha…
New study shows tradeoffs between compression quality, distortion, and perception.
problem Optimizing compression for low distortion often sacrifices perceptual quality.
method Adopted Blau & Michaeli's perceptual quality definition and studied the rate-distortion-perception tradeoff.
result Restricting perceptual quality to high generally requires a trade-off between rate and distortion.
The distortion of a curve is the supremum, taken over distinct pairs of points of the curve, of the ratio of arclength to spatial distance between the points. Gromov asked in 1981 whether a curve in every knot type can be constructed with distortion less than a universal constant C. Answering Gromov's question seems to…
Neural Estimation of the Rate-Distortion Function With Applications to Operational Source Codingcs.IT
Estimates rate-distortion function for large datasets using neural networks.
problem Designing lossy data compression schemes and comparing them with theoretical limits.
method Re-formulate rate-distortion objective and solve using neural networks.
result NERD accurately estimates the rate-distortion function for real-world datasets.
Forward Rank-Dependent Performance Criteria: Time-Consistent Investment Under Probability Distortionq-fin.MF
Introduces new performance criteria for investment under distorted probabilities.
problem Reconciling time-consistent performance with probability distortions.
method Two definitions of forward rank-dependent criteria, equivalence established; characterization of viable probability distortion processes.
result Characterization of optimal wealth process and new distorted measure.
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on how many projections are needed to accurately preserve the geometry of these man…
New coding theorem shows achievable rate matches theoretical limit.
problem Unknown existence of encoders and decoders for RDPF.
method Used stochastic, variable-length codes to prove RDPF achievable.
result Achievable rate matches theoretical rate-distortion-perception function.
Paper proposes a new black-box attack approach to minimize visual distortion.
problem Constructing adversarial examples that minimize visual distortion in a black-box threat model.
method Learning the noise distribution of adversarial examples to approximate the gradient of a non-differentiable loss function.
result The proposed attack results in much lower visual distortion compared to state-of-the-art black-box attacks.