Robustifies tree learning algorithms for corrupted data.
problem Learning latent tree structures with corrupted vector observations.
method Presented robustified algorithms using truncated inner product.
result Optimalities of robust CLRG and NJ verified by sample complexities and impossibility results.
SNJ recovers latent tree models from similarity matrices.
problem Reconstructing latent tree models from observed data.
method Spectral Neighbor Joining (SNJ) method.
result SNJ is consistent and requires fewer samples for accurate tree recovery.
New model addresses instability in hierarchical clustering of social networks.
problem Instability in existing hierarchical clustering algorithms for social networks.
method Introduce T-Stochastic Graphs, a probabilistic model that relaxes ultrametric assumptions. result Prove spectral approach combining Neighbor-Joining is statistically consistent.
We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tre…
Merging datasets is a key operation for data analytics. A frequent requirement for merging is joining across columns that have different surface forms for the same entity (e.g., the name of a person might be represented as "Douglas Adams" or "Adams, Douglas"). Similarly, ontology alignment can require recognizing disti…
Study optimal transport for stationary processes, estimating joinings and costs.
problem Optimal transport for stationary stochastic processes.
method Introduced estimators for optimal joinings and costs, established consistency and error rates.
result Consistent estimators of optimal joinings and costs under mild and stronger mixing assumptions.
Smooth SE structures on Sasaki-joins and Bott orbifolds constructed.
problem Constructing smooth Sasaki-Einstein structures on Sasaki-joins.
method Categorical relationship between Sasaki-joins and Bott orbifolds, explicit construction.
result Infinitely many smooth SE structures up to dimension eleven, conjecture for all odd dimensions.
Exploring Sasaki metrics in joined manifolds.
problem Existence of constant scalar curvature Sasaki metrics in joined manifolds.
method Investigating the Sasaki cone of joined manifolds and considering continuous families of extremal Sasaki twins.
result Conditions for the existence of constant scalar curvature Sasaki metrics in joined manifolds.
New findings on embedding simplicial complexes, showing instability under joins.
problem Conditions for embedding simplicial complexes into double dimension.
method Study of van Kampen obstructions and Smith classes.
result Smith index is not stable under joins, leading to new embeddability results.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
S. Parsa proved embedding conditions for simplicial joins.
problem Embeddability conditions for simplicial joins.
method Constructing specific simplicial complexes K and L. result The join K∗L embeds into R2(k+l+1). Extending work of Kapouleas and Yang, for any integers N≥2, k,ℓ≥1, and m sufficiently large, we apply gluing methods to construct in the round 3-sphere a closed embedded minimal surface that has genus kℓm2(N−1)+1 and is invariant under a Dkm×Dℓm subgroup of O(4), where …
Proofs show embedding conditions for complex joins and factors.
problem Embeddability conditions for complex joins and factors.
method Configuration spaces, equivariant suspension theorem, join and cone properties.
result Embeddability conditions for K∗[3] and K. Paper shows how to join Milnor fibers of real analytic functions.
problem Understanding the topology of real analytic singularities.
method Analyzes tubular Milnor fibers and their homotopy equivalence.
result Tubular Milnor fibers of f_1 + f_2 are homotopy equivalent to the join of f_1 and f_2.
We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Müller on these graphs and how to repair this lemma.
Traditional anatomical analyses captured only a fraction of real phenomic information. Here, we apply deep learning to quantify total phenotypic similarity across 2468 butterfly photographs, covering 38 subspecies from the polymorphic mimicry complex of Heliconius erato and Heliconius melpomene. E…
A new method joins two arcs with a degree of freedom.
problem Joining two arcs with a precise point.
method Geometric approach using tangent vectors and points.
result A novel method to determine the join point.
We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete,…
In this paper we show that the matrix of chromatic joins and the Gram matrix of the Temperley-Lieb algebra are similar (after rescaling), with the change of basis given by diagonal matrices.
We give a partial characterization of bordered Floer homology in terms of sutured Floer homology. The bordered algebra and modules are direct sums of certain sutured Floer complexes. The algebra multiplication and algebra action correspond to a new gluing map on SFH. It is defined algebraically, and is a special case o…
This work defines a categorical notion of principal bundles.
problem Different definitions of principal bundles in various categories.
method Formulated in join-restriction categories, which generalize partial maps.
result Shows the tangent bundle as the product of tangent space and group object.
We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X. Classical concepts known for H…
Refines Hurwitz numbers with a two-parameter theory.
problem Understanding polynomial structure and tropicalization of b-Hurwitz numbers. method Introducing CJT-refinement of symmetric functions on Fock space.
result Derives tropicalization of b-Hurwitz numbers and solves an open problem. Holomorphic cylinders converge to disks joined by flow lines.
problem Convergence of perturbed holomorphic cylinders.
method Exponential estimates and flow line computation.
result Holomorphic cylinders converge to two disks joined by a flow line.
Generalizes Sasaki join construction for quasi-regular Sasakian structures.
problem Generalize Sasaki join construction for quasi-regular Sasakian structures.
method Inductive procedure for constructing Sasakian metrics of constant scalar curvature.
result Infinitely many smooth 7-manifolds with constant scalar curvature Sasaki metrics.
We give a survey of our recent work describing a method which combines the Sasaki join construction with the admissible Kähler construction of to obtain new extremal and new constant scalar curvature Sasaki metrics, including Sasaki-Einstein metrics. The constant scalar curvature Sasaki metrics also provide explicit so…
Let H,K be two finitely generated subgroups of a free group, let ⟨H,K⟩ denote the subgroup generated by H,K, called the join of H,K, and let neither of H, K have finite index in ⟨H,K⟩. We prove the existence of an epimorphism ζ:⟨H,K⟩→F2, where F2 …
New proof shows non-embeddable polyhedra and conditions for embedding products.
problem Non-embeddability of polyhedra in high-dimensional spaces.
method Geometric cohomology and embedding conditions for polyhedra products.
result Compact polyhedra can embed in higher dimensions under specific conditions.
Study shows how to detect representation extendability using conformal measures.
problem Detecting extendability of representations using conformal measures.
method Using higher rank conformal measures and self-joinings of groups.
result Affirmative answer to detect extendability of representations.
The paper generalizes the Hausdorff dimension of limit sets for self-joinings of hyperbolic groups.
problem Calculating the Hausdorff dimension of limit sets for self-joinings of hyperbolic groups.
method The paper generalizes a classical result by considering self-joinings of convex cocompact groups and proving new inequalities for the Hausdorff dimension of directional limit sets.
result For k≤3, the paper establishes bounds on the Hausdorff dimension of directional limit sets for self-joinings of convex cocompact groups. We describe various constructions in Sasakian geometry. First we generalize the join construction of the first two authors to arbitrary Sasakian manifolds. We then give several examples, including ones which prove the existence of Sasakian-Einstein metrics on manifolds homeomorphic to S2×S5. Then we use a gen…
We propose a primitive called PJOIN, for "predictive join," which combines and extends the operations JOIN and LINK, which Valiant proposed as the basis of a computational theory of cortex. We show that PJOIN can be implemented in Valiant's model. We also show that, using PJOIN, certain reasonably complex learning and …
Characterizes homology d-manifolds with g2=3 for d≥3.
problem Classify homology d-manifolds with specific g2 values.
method Combinatorial characterization and operations like joins, retriangulations, and connected sums.
result Homology d-manifolds with g2=3 are spheres and can be derived from previous ones.
By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from Kähler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we e…
Let S be an n-punctured sphere, with n≥3. We prove that (3n) is the maximum size of a family of pairwise non-homotopic simple arcs on S joining a fixed pair of distinct punctures of S and pairwise intersecting at most twice. On the way, we show that a square annular diagram A has a corner on …
Study counts and equidistributes tori in Kleinian group self-joinings.
problem Counting and equidistribution of tori in Kleinian group self-joinings.
method Analyzes d-dimensional torus packings invariant under a self-joining of a Kleinian group. result Equidistribution results for tori with small volume in a class of d-dimensional torus packings. The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.
problem Proving acylindrical hyperbolicity for Artin groups of infinite type not associated with joins or cones.
method Developing and extending the clique-cube complex and action studies of Charney and Morris-Wright.
result Acylindrical hyperbolicity demonstrated for Artin groups of infinite type associated with graphs that are not cones.
This paper studies posets associated with link diagrams and their algebraic properties.
problem Understanding the algebraic structure of posets derived from link diagrams.
method Associaed posets with link diagrams, proved distributivity, and described join irreducibles.
result Posets of Kauffman states are distributive lattices and isomorphic to coefficient quiver posets.
Globally hyperbolic spacetimes admitting infinitely many causal (and timelike) homotopy classes of curves joining two prescribed points, are exhibited and discussed.
LNPE enhances local connections in embeddings using extended neighbor propagation.
problem Improving local connections and interactions in nonlinear dimensionality reduction.
method Inspired by GCN, LNPE extends 1-hop neighbors to n-hop neighbors in LLE.
result LNPE produces more faithful and robust embeddings with better topological and geometrical properties.
New Sasaki metrics with constant scalar curvature on sphere bundles are constructed.
problem Constructing extremal Sasaki metrics with constant scalar curvature.
method Using the fiber join construction and a recent existence theorem for constant scalar curvature Sasaki metrics.
result Explicit constructions of constant scalar curvature Sasaki metrics on specific sphere bundles.
The study shows how to construct d-spheres from (d−1)-spheres and d-balls without additional vertices.
problem Constructing d-spheres from (d−1)-spheres and d-balls without additional vertices. method Examining specific types of spheres (flag, stacked, join of spheres) and d-balls to determine if constructions can be made without extra vertices. result Affirmative answers to constructing d-spheres from (d−1)-spheres and d-balls without additional vertices for certain types of spheres and d-balls. Let H and K be subgroups of a free group of ranks h and k \geq h. We prove the following strong form of Burns' inequality: rank(H \cap K) - 1 \leq 2(h-1)(k-1) - (h-1)(rank(H \vee K) -1). A corollary of this, also obtained by L. Louder and D. B. McReynolds, has been used by M. Culler and P. Shalen to obtain information …
Study nearest-neighbor radii under dependent sampling, finding they remain informative.
problem Analyzing nearest-neighbor radii under dependent sampling.
method Consider strong mixing dependent observations, establish distribution-free almost sure convergence and sharp non-asymptotic moment bounds.
result Nearest-neighbor geometry remains informative under dependence sampling.
Deep nearest neighbors outperform self-supervised methods in anomaly detection.
problem Anomaly detection using self-supervised deep methods.
method Simple nearest-neighbor approach on Imagenet pretrained features.
result Nearest-neighbor method outperforms self-supervised methods in accuracy, few shot generalization, training time, and noise robustness.
Characterizes Lebesgue points using nearest neighbor methods.
problem Consistency of classification algorithms based on nearest neighbors.
method Characterization of Lebesgue points via 1-Nearest Neighbor regression.
result Proves convergence of 1-Nearest Neighbor classification algorithms in metric spaces.
The paper explores generalizations of Mirzakhani's recursion and computes volumes for physical gravity models.
problem Computing volumes for physical gravity models.
method Topological recursion and physical two-dimensional gravity models.
result Derivation of Virasoro constraints and cut-and-join equations for generalized Mirzakhani's recursions.
The paper studies Sasakian geometry on sphere bundles, focusing on extremal metrics and cohomology.
problem Understanding extremal and constant scalar curvature Sasaki metrics on sphere bundles.
method Applying the fiber join construction to K-contact manifolds, focusing on integral Kähler classes.
result Found infinite families of new inequivalent cone indecomposable Sasaki contact CR structures with extremal metrics.