The paper introduces a method for creating short proofs of predictions in AI models.
problem Creating reliable and explainable AI predictions.
method Defining robust hollow star numbers and analyzing certificate sizes for various hypothesis classes.
result The certificate coefficient ε x \varepsilon_x ε x precisely controls the sample size needed for predictions. Paper solves uniqueness of Steklov spectrum on specific manifolds.
problem Determining warping function from Steklov spectrum on hollow sphere manifolds.
method Proves uniqueness of warping function from Steklov spectrum.
result Warping function is uniquely determined by Steklov spectrum.
Hollow-tree Super resolves feature importance in large datasets.
problem Lack of effective scaling for large feature numbers in boosted tree models.
method Hollow-tree Super (HOTS) methodology for feature importance visualization.
result HOTS effectively resolves feature importance and directionality in high-dimensional neuroscientific data.
Estimates how many times a star appears due to gravitational lensing.
problem Estimating the number of times an observer sees a star due to gravitational lensing.
method Use affine linking numbers to estimate the number of times an observer sees a star.
result Estimates the number of times an observer sees a star due to gravitational lensing.
New star-shaped acceptability indexes generalize existing methods.
problem Generalizing existing acceptability measures.
method Characterizing acceptability indexes through star-shaped risk measures and sets.
result Introducing concrete examples linked to various financial measures.
Study tunnel numbers of cable knots and their companions, proving new bounds and constructing examples.
problem Understanding the relationship between the tunnel numbers of a knot and its cable.
method Combinatorial techniques and analysis of Heegaard splittings.
result Proves that for many cases, the tunnel number of a cable knot equals the original knot's tunnel number plus one.
The study counts geodesics on curved surfaces with specific intersections.
problem Counting geodesics with exact intersection numbers on curved surfaces.
method Introduced a dynamical scattering operator and used Pollicott-Ruelle resonances.
result Asymptotic growth of geodesics with prescribed intersections.
This paper achieves first-order regret bounds in reinforcement learning with large state spaces.
problem Achieving first-order regret bounds in reinforcement learning with large state spaces.
method Developed a novel robust self-normalized concentration bound based on the robust Catoni mean estimator.
result Obtained regret bounds scaling as O ~ ( d 3 H 3 ⋅ V 1 ⋆ ⋅ K + d 3.5 H 3 log K ) \widetilde{\mathcal{O}}(\sqrt{d^3 H^3 \cdot V_1^\star \cdot K} + d^{3.5}H^3\log K ) O ( d 3 H 3 ⋅ V 1 ⋆ ⋅ K + d 3.5 H 3 log K ) . Authors prove a formula relating the Gaussian curvature of polyhedral vertex stars to their Gauss images.
problem Proving a formula connecting discrete Gaussian curvature to the algebraic area of Gauss images.
method Comparing winding numbers and critical point index of a normal vector to deduce the formula.
result Formula significantly limits possible shapes of Gauss images of polyhedral vertex stars.
We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm's memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-…
Paper solves Serrin problem for ring-shaped domains, showing velocity has finitely many maxima.
problem Characterizing rotationally symmetric solutions to a specific PDE on a ring-shaped domain.
method Introduced new arguments in the spirit of comparison geometry to overcome the lack of monotonicity.
result Simplest conditions are not sufficient; rotational symmetry requires finitely many maxima.
Paper characterizes star-shaped risk measures and their properties.
problem Characterizing risk measures in the presence of liquidity risk and competitive delegation.
method Characterization of star-shaped risk measures, study of their properties.
result Star-shaped risk measures include all practically used risk measures.
New algorithm minimizes regret in sparse reinforcement learning.
problem Sparse reinforcement learning with unknown sparsity.
method Doubly robust approach combining feature vectors of all actions and novel analysis.
result Regret bound of i l d e O ( σ min − 1 s ⋆ H N ) ilde{O}(σ^{-1}_{\min} s_{\star} H \sqrt{N}) i l d e O ( σ m i n − 1 s ⋆ H N ) . New algorithm for robust density estimation in corrupted data.
problem Density estimation in the presence of adversarial corruption.
method Proposes an algorithm for constructing a density estimator within a star-shaped density class, derived minimax bounds for estimation.
result Obtained minimax upper and lower bounds for density estimation under adversarial corruption.
New operators help focus on specific areas in complex math problems.
problem Concentration in complex mathematical structures.
method Construct conjugate-linear perturbations of twisted spinc Dirac operators using the conjugate-linear Hodge star operator.
result These perturbations satisfy the concentration principle.
We introduce a method to design lightweight shell objects that are structurally robust under the external forces they may experience during use. Given an input 3D model and a general description of the external forces, our algorithm generates a structurally-sound minimum weight shell object. Our approach works by alter…
New method robust to semi-random sparse recovery, nearly-linear time.
problem Brittleness of fast sparse recovery algorithms under generative model changes.
method Designing a new iterative method robust to semi-random model.
result Proves robustness of new method to semi-random generative models.
The paper defines new polynomials for links and linkoids.
problem No specific problem stated; focus on new polynomials.
method Defined as sums over states of link or linkoid diagrams with f = n f=n f = n . result Constructed new polynomials for starred links and linkoids.
Develops an ℓ_p theory for PCA and spectral clustering.
problem Lack of precise characterizations of PCA scores for low-dimensional embedding.
method An ℓ_p perturbation theory for PCA in Hilbert spaces, analyzing eigenvectors and Gram matrix.
result Optimal recovery results for Gaussian mixture and stochastic block models.
We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with dimension-independent error guarantees for several families of structured distributions…
Proposes an efficient alternative to nonconvex-nonconcave min-max optimization.
problem Min-max optimization challenges in nonconvex-nonconcave settings.
method Introduces ε-greedy adversarial equilibrium model and proves its existence.
result Existence of ε-greedy adversarial equilibrium for smooth bounded functions.
The study shows infinitely many Reeb orbits on star-shaped hypersurfaces with growth rate like prime numbers.
problem Growth rate of Reeb orbits on star-shaped hypersurfaces.
method Analyzing fiberwise star-shaped hypersurfaces in cotangent bundles with topological conditions.
result The number of Reeb orbits with period at most T grows at least like T/log(T).
We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation X \mathfrak{X} X of the sum of an approximately) low rank matrix Θ ⋆ Θ^\star Θ ⋆ with a second matrix Γ ⋆ Γ^\star Γ ⋆ endowed with a complementary …
We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the complement of "link" in sphere are equivalents, since they exist. Moreover, in ord…
Let c be a periodic Reeb orbit on the boundary S of a compact star-shaped domain C in R4. We show that if there is an immersed symplectic disc f in C with boundary c then the self-linking number lk(c) of c equals 2 tan(f)-1 where tan(f) is the tangential self-intersection number of f. We also show that if C is convex a…
Let { ⋅ , ⋅ } P \{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}} { ⋅ , ⋅ } P be a variational Poisson bracket in a field model on an affine bundle π π π over an affine base manifold M m M^m M m . Denote by × \times × the commutative associative multiplication in the Poisson algebra A \boldsymbol{\mathcal{A}} A of local functionals Γ ( π ) → k Γ(π)\to\Bbbk Γ ( π ) → k that take…
Improved guarantees for nonconvex matrix factorization with rank overparameterization.
problem Minimizing nonconvex objective over low-rank matrices.
method Overparameterized Burer--Monteiro approach, leveraging smoothness and strong convexity.
result Local optimization globally converges to global optimum under certain rank conditions.
The paper explores graphons of line graphs from sparse finite graphs.
problem Estimating graph limits from sparse finite graphs.
method Mapping finite graphs to their line graphs and analyzing graphs with the square-degree property.
result Graphons of line graphs can distinguish between sparse graphs like star graphs and superlinear preferential attachment graphs.
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.
The study proves that in normal tilings, at least two vertices are required per cell.
problem Understanding the minimum number of vertices required in normal tilings.
method The research examines both periodic and monohedral tilings in 2D, proving the minimum number of non-smooth vertices required.
result The study confirms that for normal tilings, at least two vertices are necessary per cell.
The goal of this study is to present the development of a machine learning based approach that utilizes phase space alone to separate the Gaia DR2 stars into two categories: those accreted onto the Milky Way from those that are in situ. Traditional selection methods that have been used to identify accreted stars typica…
We calculated the cross correlations between the half-hourly times series of the ten Dow Jones US economic sectors over the period February 2000 to August 2008, the two-year intervals 2002--2003, 2004--2005, 2008--2009, and also over 11 segments within the present financial crisis, to construct minimal spanning trees (…
Algorithm achieves optimal regret for unknown Lipschitz convex losses.
problem Online learning with unknown Lipschitz constant and target vector norm.
method Develops an online learning algorithm without knowledge of G G G or ∥ w ⋆ ∥ \|w_\star\| ∥ w ⋆ ∥ . result Matches optimal regret bound G ∥ w ⋆ ∥ T G\|w_\star\|\sqrt{T} G ∥ w ⋆ ∥ T up to logarithmic factors. Minimizing a convex, quadratic objective of the form f A , b ( x ) : = 1 2 x ⊤ A x − ⟨ b , x ⟩ f_{\mathbf{A},\mathbf{b}}(x) := \frac{1}{2}x^\top \mathbf{A} x - \langle \mathbf{b}, x \rangle f A , b ( x ) := 2 1 x ⊤ A x − ⟨ b , x ⟩ for A ≻ 0 \mathbf{A} \succ 0 A ≻ 0 is a fundamental problem in machine learning and optimization. In this work, we prove gradient-query complexity lower bounds for minimizing conv…
Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.
problem What conditions guarantee global existence of Gage's area-preserving flow for nonconvex initial curves?
method Using Dittberner's singularity analysis theory, constructed a ``flying wing'' curve to show limitations.
result Gage's area-preserving flow does not always preserve star-shapedness of evolving curves.
New algorithm improves heteroskedastic PCA performance.
problem Estimating low-rank matrix subspace from noisy data.
method Deflated-HeteroPCA algorithm, dividing spectrum into subblocks.
result Near-optimal and condition-number-free statistical guarantees.
In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric Q m Q^m Q m . It is proved that there exist no Hopf hypersurfaces in Q m , m ≥ 3 Q^m,m\geq3 Q m , m ≥ 3 , with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on M M M …
Optimal scheme minimizes deviation in federated transfer learning for kernel regression.
problem Minimizing cumulative deviation in federated transfer learning across multiple datasets.
method Regret-optimal iterative scheme for continual communication between nodes and server.
result Explicit updates for the regret-optimal algorithm in finite-rank kernel regression.
Classifies star products on Lie algebroid duals and extends to projectable quantizations.
problem Classifying and extending quantizations on Lie algebroid duals.
method Classification through second Lie algebroid cohomology, extension to projectable quantizations.
result Quantization commutes with reduction in the considered setting.
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in Z / a Z \mathbb{Z}/a\mathbb{Z} Z / a Z . We introduce a congruence level definition and a property of a primitive t…
Flow turns star-shaped curves into circles.
problem Transforming star-shaped curves into circles.
method Gage's area-preserving flow.
result Curves evolve into circles over time.
The paper studies dynamic star-shaped risk measures and their representation.
problem Representing dynamic star-shaped risk measures and their properties.
method Representation theorems for dynamic monetary and star-shaped risk measures.
result Dynamic star-shaped risk measures can be represented as the lower envelope of a family of dynamic convex risk measures.
Paper disproves symmetry of stars at infinity in a specific graph.
problem Symmetry of stars at infinity in a specific graph.
method Defined incidence geometry of stars at infinity; provided an example.
result Relation of one boundary point being included in a star of another is not symmetric.
Study star products on Poisson manifolds compatible with reduction.
problem Finding star products compatible with coisotropic reduction.
method Compute second constraint Hochschild cohomology of constraint algebra.
result Determine infinitesimal star products on Poisson manifolds.
The paper characterizes dynamic return and star-shaped risk measures via BSDEs.
problem Characterizing dynamic return and star-shaped risk measures.
method Characterization of star-shaped functionals and BSDEs.
result Existence of convex BSDEs with non-empty set of supersolutions.
Gradient descent learns over-param neural nets better than NTK.
problem Learning over-parametrized neural networks with ReLU activations.
method Gradient descent from random initialization on a Gaussian input distribution.
result Gradient descent achieves population loss o ( 1 / d ) o(1/d) o ( 1/ d ) , while NTK achieves Ω ( 1 / d ) Ω(1/d) Ω ( 1/ d ) . Deform moment map on symplectic connections using star product algebras.
problem Understanding symplectic connections and their deformations.
method Study vector bundle of Fedosov star product algebras, formal connection, curvature, and star product trace.
result Showed star product trace as a formal symplectic form and moment map.