arXiv research

Vertex distortion detects if a knot is unknot.

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…

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…

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $π/2$ is…

In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial $μ$-entropy. Our basic principle tells that once correctly renormalized, the metric-measure quantities obey similar estimates as in the non-collapsing case…

The study finds minimal distortion embeddings of surfaces into small domains.

Paper studies fundamental limits of communication in distributed learning.

We show that the distortion of the (2,q)-torus knot is not bounded linearly from below.

Sharp bounds for distortion risk metrics under uncertain distributions.

New bounds on generalization error for distributed learning using rate-distortion theory.

Our main result is a nontrivial lower bound for the distortion of some specific knots. In particular, we show that the distortion of the torus knot $T_{p,q}$ satisfies $δ(T_{p,q})>\frac 1{160}\min(p,q)$. This answers a 1983 question of Gromov.

Triangulates surfaces with bounded energy using diffeomorphisms.

Paper quantifies distortion risk measures' robustness to distributional uncertainty.

Study bounds on curvature for special Finsler metrics.

Lower bounds on Bayes risk for realizable models derived using information theory.

New bounds for optimal transport using Gaussian processes and rate-distortion functions.

New coding theorem shows achievable rate matches theoretical limit.

We prove that distortion of a knotted curve in $\R^3$ is great than 4.76. This improves a result obtained by John M. Sullivan and Elizabeth Denne in \cite{DS}.

New bounds on knot distortion and Seifert surface properties.

We present an information-theoretic framework for bounding the number of labeled samples needed to train a classifier in a parametric Bayesian setting. We derive bounds on the average $L_p$ distance between the learned classifier and the true maximum a posteriori classifier, which are well-established surrogates for th…

Paper introduces new risk measures that unify two existing types.

Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer CAV17]. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available m…

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 $d$ mapping class group is the same as that o…

We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ 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…

ReLU networks don't exponentially distort curve lengths as previously thought.

We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unb…

Unified bounds linking compressibility, fractal dimensions, and mutual information.

We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares intrinsic distances in leaves with extrinsic distances in the ambient space grows fast…

Motivated by models of human decision making proposed to explain commonly observed deviations from conventional expected value preferences, we formulate two stochastic multi-armed bandit problems with distorted probabilities on the reward distributions: the classic $K$-armed bandit and the linearly parameterized bandit…

Distortion risk measures are extensively used in finance and insurance applications because of their appealing properties. We present three methods to construct new class of distortion functions and measures. The approach involves the composting methods, the mixing methods and the approach that based on the theory of c…

The paper analyzes extreme risk measures with limited distributional information.

In this article we use rate-distortion theory, a branch of information theory devoted to the problem of lossy compression, to shed light on an important problem in latent variable modeling of data: is there room to improve the model? One way to address this question is to find an upper bound on the probability (equival…

Given a metric space $X$ and a function $f: X \to \mathbb{R}$, the Reeb construction gives metric a space $X_f$ together with a quotient map $X \to X_f$. Under suitable conditions $X_f$ becomes a metric graph and can therefore be used as a graph approximation to $X$. The Gromov-Hausdorff distance from $X_f$ to $X$ is b…

We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(S^n), thought of as a discrete group. An appendix by Y …

The paper calculates bounds for risk metrics and entropies under partial information constraints.

It is well-known that quasi-isometries between R-trees induce power quasi-symmetric homeomorphisms between their ultrametric end spaces. This paper investigates power quasi-symmetric homeomorphisms between bounded, complete, uniformly perfect, ultrametric spaces (i.e., those ultrametric spaces arising up to similarity …

Given a space $Y$ in $X$, a cycle in $Y$ may be filled with a chain in two ways: either by restricting the chain to $Y$ or by allowing it to be anywhere in $X$. When the pair $(G,H)$ acts on $(X, Y)$, we define the $k$-volume distortion function of $H$ in $G$ to measure the large-scale difference between the volumes of…

We discuss boundedness and distortion in transformation groups. We show that the groups $\mathrm{Diff}^r_0(\mathbb{R}^n)$ and $\mathrm{Diff}^r(\mathbb{R}^n)$ have the strong distortion property, whenever $0 \leq r \leq \infty, r \neq n+1$. This implies in particular that every abstract length function on these groups i…

Paper proposes robust risk measures for non-negative risks with partial information.

This paper develops a method to estimate the rate-distortion function for general data sources.

Researchers prove inner product recovery is impossible in latent space models.

Simplifies VAE for anomaly detection using rate-distortion theory.

The paper tightens bounds on distances between Reeb graphs.

New method estimates rate-distortion function using optimal transport.

The paper connects geometric and topological concepts to bound distances between metric spaces.

Maps persistence diagrams into Hilbert and Euclidean spaces with explicit distortions.

Algorithm finds optimal affine transformation to minimize overall distortion.

We show that a closed, connected, oriented, Riemannian $n$-manifold, admitting a branched cover of bounded length distortion from $\mathbb R^n$, has a virtually Abelian fundamental group.