arXiv research

Paper discusses natural quasiconvexity and its relation to decomposable sums in risk measures.

By applying the lantern relation substitutions to the positive relation of the genus two Lefschetz fibration over $\mathbb{S}^{2}$. We show that $K3\#2 \overline{\mathbb{CP}}{}^{2}$ can be rationally blown down along seven disjoint copies of the configuration $C_2$. We compute the Seiberg-Witten invariant of the result…

Proves special alternating knots cannot be decomposed as non-trivial band sums.

Decomposable arrangements have simpler topological and combinatorial properties.

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish in [3]; Lickorish proved [5] that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produc…

The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.

The Möbius energy is one of the knot energies, and is named after its Möbius invariant property. It is known to have several different expressions. One is in terms of the cosine of conformal angle, and is called the cosine formula. Another is the decomposition into Möbius invariant parts, called the decomposed Möbius e…

The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fai…

Given $(V_1,V_2)$ a Heegaard splitting of the complement of a composite knot $K=K_1# K_2$ in $S^3$, where $K_i, i=1,2$ are prime knots, we have a unique, up to isotopy, decomposing annulus $A$. When the intersection of $A$ and $V_1$ is a minimal collection of disks we study the components of $V_1-N(A)$ and show that at…

Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to $n^{\lfloor p/2 \rfloor}$ for a $p$-th order tensor in $\mathbb{R}^{n^p}$. Previously no efficient algorithm can decompose 3rd order ten…

Differentially private algorithms for submodular maximization under various constraints.

Study shows instantons and monopoles decompose into U(1) components.

Divergence functions play a key role as to measure the discrepancy between two points in the field of machine learning, statistics and signal processing. Well-known divergences are the Bregman divergences, the Jensen divergences and the f-divergences. In this paper, we show that the symmetric Bregman divergence can be …

We use results of Matzeu and Nikcevic to decompose the space of affine Kaehler curvature tensors as a direct sum of irreducible modules in the complex setting

This work classifies belted sum decompositions of fully augmented links.

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold X_K becoming diffeomorphic to X after a single stabilization by connected summing with S^2 …

In this article we prove that, if $X$ is a smooth $4$-manifold containing an embedded double node neighborhood, all knot surgery $4$-manifolds $X_K$ are mutually diffeomorphic to each other after a connected sum with $\mathbb{CP}^2$. Hence, by applying to the simply connected elliptic surface $E(n)$, we also show that …

We examine the relationship between the (untwisted) knot Floer cube of resolutions and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot $K$, we see that the filtered complex decomposes as a direct sum of HOMFLY-PT homologies of various subdiagrams. Jaeger's c…

The quotients $Y=X/conj$ by the complex conjugation $conj\: X\to X$ for complex rational and Enriques surfaces $X$ defined over $\R$ are shown to be diffeomorphic to connected sums of $\barCP2$, whenever $Y$ are simply connected.

Let (M, g, omega) be a compact, almost-Kaehler Einstein 4-manifold of negative star-scalar curvature. Then (M, omega) is a MINIMAL symplectic 4-manifold of general type. In particular, M cannot be differentiably decomposed as a connected sum N # (-CP_2).

We propose an algorithm for a family of optimization problems where the objective can be decomposed as a sum of functions with monotonicity properties. The motivating problem is optimization of hyperparameters of machine learning algorithms, where we argue that the objective, validation error, can be decomposed as mono…

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, th…

We study the converse to the statement that instantons are minimizers of the Yang--Mills energy in four dimensions. We show that given an energy minimizing connection, A, the curvature of A takes values in a subbundle of the adjoint bundle which decomposes as a sum of instantons.

We analyze symplectic forms on six dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable solvable algebras with a four dimensional nilradical.

Spatial graphs are decomposed into planar forests and braids.

Knots can be constructed and decomposed using Murasugi sums of Seifert surfaces.

Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.

Sum-product networks have recently emerged as an attractive representation due to their dual view as a special type of deep neural network with clear semantics and a special type of probabilistic graphical model for which inference is always tractable. Those properties follow from some conditions (i.e., completeness an…

We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the Kähler classes in dense subse…

Let $P(E)$ be the projectivization of a holomorphic vector bundle $E$ over a compact complex curve $C$. We characterize the existence of an extremal Kähler metric on the ruled manifold $P(E)$ in terms of relative K-polystability and the fact that $E$ decomposes as a direct sum of stable bundles.

We decompose the de Rham Laplacian on Sasaki-Einstein manifolds as a sum over mostly positive definite terms. An immediate consequence are lower bounds on its spectrum. These bounds constitute a supergravity equivalent of the unitarity bounds in dual superconformal field theories. The proof uses a generalization of Kah…

Resolution of a compact group action in the sense described by Albin and Melrose is applied to the conjugation action by the unitary group on self-adjoint matrices. It is shown that the eigenvalues are smooth on the resolved space and that the trivial bundle smoothly decomposes into the direct sum of global one-dimensi…

We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy exampl…

The main theorem describes the behaviour of the stable cohomotopy invariant defined in the first article (joint with M. Furuta) in this series of two under the operation of taking connected sums of four-manifolds: The invariant of a connected sum is the smash product (in the sense of equivariant spectra) of the invaria…

Analyzes first exit times in a modified Barndorff-Nielsen and Shephard model.

The study preserves positive Ricci curvature on connected sums of fibre bundles.

We study the microlocal properties of the geodesic X-ray transform $\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \math…

We present a novel tractable generative model that extends Sum-Product Networks (SPNs) and significantly boosts their power. We call it Sum-Product-Quotient Networks (SPQNs), whose core concept is to incorporate conditional distributions into the model by direct computation using quotient nodes, e.g. $P(A|B) = \frac{P(…

A new method decomposes Bayesian uncertainty into per-class contributions for safer classification.

Decomposes epistemic uncertainty into per-class contributions for safer classification.

Diagonal metrics solve Hermitian-Einstein equations for decomposed Higgs bundles.

Researchers classify and decompose valuations on convex functions.

The study proves manifolds with positive scalar curvature can be decomposed into spherical and toroidal pieces.

Paper tackles noisy labels for non-decomposable performance measures.

The paper constructs quantum invariants for knotoid diagrams.

We study consistency of learning algorithms for a multi-class performance metric that is a non-decomposable function of the confusion matrix of a classifier and cannot be expressed as a sum of losses on individual data points; examples of such performance metrics include the macro F-measure popular in information retri…

Study private submodular maximization in streaming data.

We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding Lovász extensions and the squared Euclidean norm, leading to algorithms requiring …