arXiv research

Heegaard Floer homology study confirms composition maps match up to homotopy.

The study reveals simplicity bias in neural networks leading to better compositional mappings.

We consider multi-level composite optimization problems where each mapping in the composition is the expectation over a family of random smooth mappings or the sum of some finite number of smooth mappings. We present a normalized proximal approximate gradient (NPAG) method where the approximate gradients are obtained v…

We consider the problem of minimizing the composition of a smooth (nonconvex) function and a smooth vector mapping, where the inner mapping is in the form of an expectation over some random variable or a finite sum. We propose a stochastic composite gradient method that employs an incremental variance-reduced estimator…

The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.

Unified tractability conditions for various compositional inference queries.

AdaGrad fails to adapt to Hölder-smoothness in composite optimization problems.

Improved subgradient method tackles ill-conditioned composite optimization problems.

New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.

New research shows Manturov-Nikonov map fails for large k values.

New mappings on closed manifolds can't be broken down easily.

Study on conditions for singular local tube fibrations.

Paper develops momentum schemes with variance reduction for non-convex composition optimization.

We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T, there is a full measure set of αin [0, 1) so that T composed with R_α is uniquely ergodic, where R_α is rotation by α.

We present a graphical criterion for reading dependencies from the minimal directed independence map G of a graphoid p when G is a polytree and p satisfies composition and weak transitivity. We prove that the criterion is sound and complete. We argue that assuming composition and weak transitivity is not too restrictiv…

Godin introduced the categories of open closed fat graphs $Fat^{oc}$ and admissible fat graphs $Fat^{ad}$ as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that $Fat^{ad}$ is a model of the mapping class group of open-close…

Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\mathbb{R}^m$). Let $f:N\to U$ and $F:U\to \mathbb{R}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$…

Floer cohomology is computed for certain elements of the mapping class group of a surface $Σ$ of genus $g>1$ which are compositions of positive and negative dehn twists along some loops in $Σ$. The computations cover a certain class of pseudo-Anasov maps.

This paper analyzes how diffusion models learn and generalize concepts.

A new geometry-preserving method for interpreting compositional data.

A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be factored as the composition of an embedding and a projection for certain dimensions.

Embedding models for entities and relations are extremely useful for recovering missing facts in a knowledge base. Intuitively, a relation can be modeled by a matrix mapping entity vectors. However, relations reside on low dimension sub-manifolds in the parameter space of arbitrary matrices---for one reason, compositio…

Paper develops polynomial approximations for complex probability densities.

The paper describes fitting submanifolds to data using Sussmann's orbit theorem.

The ERI is a new index for measuring exam readiness.

A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable…

We prove that any smooth harmonic map from $S^3$ into $S^2$ of Morse index less or equal than $4$ has to be an harmonic morphism, that is the successive composition of an isometry of $S^3$, the Hopf fibration and an holomorphic map from ${\mathbb C}P^1$ into itself.

Generalizes Newmark methods for nonholonomic systems.

A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of…

Here we study non-convex composite optimization: first, a finite-sum of smooth but non-convex functions, and second, a general function that admits a simple proximal mapping. Most research on stochastic methods for composite optimization assumes convexity or strong convexity of each function. In this paper, we extend t…

Analytic functions on specific domains are characterized by their smoothness and composites with polynomial curves.

New infinite-type loxodromic elements found in surface mapping classes.

Analyzing large X-ray diffraction (XRD) datasets is a key step in high-throughput mapping of the compositional phase diagrams of combinatorial materials libraries. Optimizing and automating this task can help accelerate the process of discovery of materials with novel and desirable properties. Here, we report a new met…

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.

Deep neural networks can approximate complex functions through repeated compositions of a fixed-size ReLU network.

We prove that, in general, given a $p$-harmonic map $F:M\to N$ and a convex function $H:N\to\mathbb{R}$, the composition $H\circ F$ is not $p$-subharmonic. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. The key of the proof is an asymptotic…

In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$ as the image of the composition of the Hopf map $\mathcal{H}: \mathbb{S}^7\to \mathbb{CP}^3$ and a map $ψ:\mathbb{R}^3 \to \mathbb{S}^7$ with certain conditions.

Geometric interpretation of 3-manifold invariants using immersed curves.

Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.

We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We gi…

There is tremendous interest in precision medicine as a means to improve patient outcomes by tailoring treatment to individual characteristics. An individualized treatment rule formalizes precision medicine as a map from patient information to a recommended treatment. A treatment rule is defined to be optimal if it max…

Cosmos models scenes using neural encodings and symbolic attributes for compositional generalization.

Algorithm counts intersections of normal curves efficiently.

Optimizes natural frequencies of cellular composites with various microstructures.

Enhances neural networks with prior knowledge through a composite kernel.

Proves geodesic connections on 2-torus without invariant tori.

Low-rank inducing unitarily invariant norms have been introduced to convexify problems with low-rank/sparsity constraint. They are the convex envelope of a unitary invariant norm and the indicator function of an upper bounding rank constraint. The most well-known member of this family is the so-called nuclear norm. To …

Generative model synthesizes complex data structures with composite and nested types.