Paper develops EP algorithm for t-exponential family using q-algebra.
problem Efficient learning algorithm for t-exponential family distributions.
method Borrowing q-algebra from statistical physics, develop EP algorithm.
result Demonstrates performance of EP algorithm on Bayes point machine and Student-t process classification.
Bayesian deep learning with flexible nonlinearities and t-exponential family.
problem Accounting for uncertainty in deep network parameters with small datasets.
method Novel deep network architectures with randomized nonlinearities, trained via approximate inference using t-divergence.
result Model achieves good test error bounds and outperforms related techniques on challenging benchmarks.
Improved language models using t-exponential distributions for sparse sequential data.
problem Sparse sequential data and complex temporal dynamics in language modeling.
method Adopted variational inference with multivariate t-exponential distributions to model latent variables.
result Improved performance on language modeling benchmarks compared to existing techniques.
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.
We introduce a numerical isomorphism invariant p(T) for any triangulation T of S^3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable then p(T) is `small' in the sense that we obtain a linear …
This paper tackles unknown causal graphs and soft interventions, establishing regret bounds and an efficient algorithm.
problem Designing causal bandit algorithms with unknown causal graphs and stochastic intervention models.
method Establishes novel regret bounds and presents a computationally efficient algorithm for unknown graph and soft interventions.
result Regret bounds for unknown graph and soft interventions, with a universal minimax lower bound.
Constructs families of Toeplitz operators for symplectic fibrations.
problem Quantization of symplectic fibrations.
method Smooth families of Szegö projections and Toeplitz operators.
result Deformation quantization of prequantizable symplectic fibrations.
In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebrai…
Proves an equivariant version of index theorem for geometric families.
problem Index theorem for geometric families with group action.
method Apply equivariance --> families principle to Clifford module bundles.
result Equivariant version of Bismut's families index theorem.
Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.
problem Developing a new class of probability models that are easier to handle and have useful properties.
method Introducing squared families as families of probability densities obtained by squaring a linear transformation of a statistic, and showing their properties and applications.
result Squared families have convenient properties and can approximate target densities well.
Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.
problem Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
method Riemannian geometry of Bergman metrics and smoothness of families of isometries.
result Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
Computes Seiberg-Witten invariants for Kähler families of 4-manifolds.
problem Computing invariants for families of Kähler 4-manifolds.
method Generalized Seiberg-Witten invariants for smooth families of 4-manifolds with Kähler structures.
result Computed invariants for specific Kähler families in terms of characteristic classes.
We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded in the standard contact (2n+1)-space (for n>1) that are contractible in the smo…
Study families of flat connections with nilpotent Higgs fields, showing similar monodromy to regular Higgs bundles.
problem Investigate Cimes-families of flat connections with nilpotent Higgs fields. method Analyze families of flat connections including real twistor lines and conformal limits, deducing monodromy similarities.
result Traces of holonomies are asymptotically exponential in rational powers of the parameter of the family.
Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We gi…
Study families of Morse functions for manifolds with boundary.
problem Characterize degeneracies in 1-parameter families of Morse functions.
method List all possible degeneracies in generic 1-parameter families.
result Identified all degeneracies in generic 1-parameter families.
We consider the local analytic behavior for a family of holomorphic differentials on a family of degenerating annuli. Three results and discussion are presented. The first is the normal families Lemma 1. The second is an isomorphism of sheaves, formula (3), giving a direct description of families of regular k-differe…
Extends width estimates to family case using index theory.
problem Sharp width estimates for Riemannian bands with positive scalar curvature.
method Dirac operators and family index theory.
result Proves width estimate for fiber bundles with infinite A-hat area.
Consider two families of closed oriented curves in a d-manifold. At each point of intersecction of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an i-dimensional family and a j-dimensional family will produce …
Paper introduces kernel deformed exponential families for sparse continuous attention.
problem Creating efficient attention mechanisms for sparse data.
method Developed kernel deformed exponential families, theoretically and experimentally.
result Kernel deformed exponential families can attend to multiple compact regions of data.
New formula connects Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds.
problem Understanding invariants of families of 4-manifolds.
method Cohomological formula linking Bauer-Furuta and Seiberg-Witten invariants.
result New divisibility properties of families Seiberg-Witten invariants.
After defining reduced minimum braid word and criteria for a braid family representative, different braid family representatives are derived, and a correspondence between them and families of knots and links given in Conway notation is established.
Study algebraic relations of Vassiliev invariants for families of knots.
problem Understanding algebraic structure of Vassiliev invariants for knot families.
method Analyzing algebraic relations and generating sets of Vassiliev invariants in 3D Chern-Simons theory.
result For 1-parametric knot families, Vassiliev invariants are finitely generated. For more parameters, there can be an infinite number of generators.
Constructs non-abelian G2-instantons on ALC members of B7 family.
problem Constructing non-abelian G2-instantons on ALC members of B7 family.
method Using co-homogeneity one symmetries, classify and describe the solutions as perturbations of abelian instantons.
result Find a one-parameter family of instantons with polynomial decay.
The paper introduces structured variational families to improve scalability in black-box variational inference.
problem Scalability issues in black-box variational inference, especially for large datasets and hierarchical models.
method Developed structured variational families that achieve better iteration complexity of O(N) compared to full-rank families.
result Structured variational families can achieve better scaling with respect to dataset size N, improving iteration complexity from O(N^2) to O(N).
Proves conditions for generating families on Lagrangian cobordisms.
problem Existence of generating families on Lagrangian cobordisms.
method Analyzes exact Lagrangian cobordisms and Legendrian submanifolds using generating families.
result Conditions for extending generating families linear at infinity.
In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…
New families of embeddings in 4-manifolds, topologically trivial but smoothly non-trivial.
problem Constructing non-trivial smooth embeddings of 3-manifolds in 4-manifolds.
method Parameterized families of embeddings, using high-dimensional spheres.
result Embeddings of homology spheres and any 3-manifold in blown-up K3 surfaces.
Characterizes level-set families of harmonic functions without critical points.
problem Understanding level-set families of harmonic functions without critical points.
method Characterization via local differential-geometric condition and construction from geometric data.
result Evolution of gradient of harmonic functions determined by mean curvature of level sets.
In this paper, we first prove a local family version of the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, then we extend the famous Witten's rigidity Theorems to the family case. Several family vanishing theorems for elliptic genera are also proved.
The paper proves rigidity of Seiberg-Witten invariants for spin 4-manifold families.
problem Rigidity of Seiberg-Witten invariants for spin 4-manifold families.
method Mechanism of rigidity theorem and 10/8-type inequality.
result Existence of non-smoothable topological families of 4-manifolds.
New 2-parameter family of triply periodic minimal surfaces, sharing properties with Schwarz' D but distinct.
problem Proving the existence of a new family of triply periodic minimal surfaces.
method Proved existence through mathematical proof and analysis of properties.
result New family oΔ does not belong to Meeks' five-dimensional family but shares properties with Schwarz' D. Algorithm finds smooth primitives for exact forms.
problem Computing smooth primitives for exact forms on manifolds.
method Diagram chasing in Čech-de Rham complex, explicit formulas.
result Explicit formulas for primitive families.
It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family to any simply connected elliptic surface for which all curvature ellipses of a …
Study circle families' envelopes and related curves.
problem Understanding relationships between circle families and special curves.
method Investigate envelopes of circle families and their connections to evolutes, pedals, evolutoids, and pedaloids.
result Characterized relationships between circle families and related curves.
Geometric equation defines canonical metrics on vector bundle families.
problem Finding canonical metrics on families of holomorphic vector bundles.
method Introducing a geometric partial differential equation for families of holomorphic vector bundles.
result Construction of Hermite--Einstein metrics in adiabatic classes on product manifolds and proof of the existence of a unique solution for the Dirichlet problem.
In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…
Survey of gauge theory for families of 4-manifolds.
problem Understanding gauge theory for families of 4-manifolds.
method Developed gauge theory for diffeomorphism groups of 4-manifolds.
result New applications of gauge theory to diffeomorphism groups.
We study the problem of recovering the subspace spanned by the first k principal components of d-dimensional data under the streaming setting, with a memory bound of O(kd). Two families of algorithms are known for this problem. The first family is based on the framework of stochastic gradient descent. Nevertheles…
The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.
problem Understanding the balance of conflict graphs in Petersen family graphs.
method Analyzing maximally planar subgraphs and their conflict graphs.
result All but three strong conflict graphs from Petersen Family Graphs are unbalanced.
We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in R3. These new examples form branches issuing from the H-family, the rPD-family, the tP-family, and the tD-family, that converge to some degenerate embedding of the families. As to nonde…
The study explores generalized divergences and exponential families with a focus on sufficient conditions and laws of large numbers.
problem Generalization of Kullback-Leibler divergence and exponential families.
method Investigation of (h,τ)-divergence and (h,τ)-exponential families, definition of (h,τ)-dependence, proof of law of large numbers. result Sufficient condition for (h,τ)-divergence to induce Hessian structure on (h,τ)-exponential family, proof of law of large numbers. An analytic index is defined for a family of cusp pseudodifferential operators, Pb, on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation Qb by a family of cusp operators of…
Solves four problems related to circle families in the plane.
problem Four basic problems of circle families in the plane.
method Solves all four basic problems of circle families in the plane.
result All four basic problems are solved.
In this paper, we give proofs of the family index formula and the equivariant family index formula by the Greiner's approach to heat kernel asymptotics. We compute equivariant family JLO characters. We also define the equivariant eta form and give a proof of its regularity.
Generalizes moment-matching for exponential families with conditioning or hidden data.
problem Generalizing moment-matching conditions for exponential families with conditioning or hidden data.
method First-principles explanation and self-contained derivation of generalized moment-matching conditions.
result Derives generalized moment-matching conditions for conditional exponential families and hidden data.
We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection as in the local family index theorem.
New polynomials defined for quandle structures, enhancing graph invariants.
problem Enhancing the counting invariant for spatial graphs and handlebody-links.
method Introducing quandle polynomials and G-family polynomials for quandles, defining enhancements for invariants.
result New enhancements of the G-family counting invariant for trivalent spatial graphs and handlebody-links.