We found a new simple family of Cantor sets whose projections are one-dimensional.
problem Finding simple Cantor sets with specific projection properties.
method Developed a new series of self-similar Cantor sets in R3. result All projections of these new Cantor sets are connected and one-dimensional.
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
We introduce a 1-cocycle on the group of diffeomorphisms Diff(M) of a smooth manifold M endowed with a projective connection. This cocycle represents a nontrivial cohomology class of $\Diff(M)$ related to the Diff(M)-modules of second order linear differential operators on M. In the one-dimensional case, this c…
New Cantor sets with high-dimensional projections discovered.
problem Understanding projections of Cantor sets in high dimensions.
method Construction and analysis of Cantor sets in Rn. result Cantor sets can be moved to have (n−2)-dimensional projections in (n−1)-planes. Estimation of low-rank matrices is of significant interest in a range of contemporary applications. In this paper, we introduce a rank-one projection model for low-rank matrix recovery and propose a constrained nuclear norm minimization method for stable recovery of low-rank matrices in the noisy case. The procedure is…
A new method steers Gaussian distributions with minimal effort.
problem Steering high-dimensional Gaussian distributions efficiently.
method Sliced feedback controller using one-dimensional projections and averaging.
result The method steers Gaussian distributions to targets efficiently.
s-OTDD compares datasets efficiently without training, robust to class variations.
problem Efficiently compare datasets without training or class variations.
method Moment Transform Projection (MTP) and sliced optimal transport.
result s-OTDD correlates with optimal transport and transfer learning performance.
A method for clustering small datasets in high dimensions using random projections.
problem Challenges in clustering small datasets in high-dimensional spaces.
method Random projection followed by binary clustering in one-dimensional space.
result Statistically significant clustering structures can be found with as few as 100-200 points.
The paper characterizes convex co-compact groups with one-dimensional boundary faces.
problem Characterizing convex co-compact groups with specific boundary properties.
method Proving relative hyperbolicity and using coarse Hilbert dimension.
result Convex co-compact groups with one-dimensional boundary faces are relatively hyperbolic.
New method estimates SW distance using CDFs for scalable data parallelism.
problem Estimating SW distance efficiently for large datasets.
method Estimators based on CDFs of projected measures, avoiding sorting.
result Efficient estimation for large datasets and federated learning.
Flexible links have symplectic representatives in complex projective space.
problem Existence of symplectic representatives for flexible links.
method Construction of a symplectic surface invariant under complex conjugation.
result No obstructions to finding symplectic representatives beyond classical topology.
Product models of low dimensional experts are a powerful way to avoid the curse of dimensionality. We present the ``under-complete product of experts' (UPoE), where each expert models a one dimensional projection of the data. The UPoE is fully tractable and may be interpreted as a parametric probabilistic model for pro…
The purpose of this paper is to classify α-para Kenmotsu manifolds M3 such that the projection of the image of concircular curvature tensor L in one-dimensional linear subspace of Tp(M3) generated by ξp is zero.
We show that a general n-dimensional polarized abelian variety (A,L) of a given polarization type and satisfying h0(A,L)≥28n⋅n!nn is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety i…
Sliced kernelized Stein discrepancy improves goodness-of-fit tests and model learning in high dimensions.
problem The curse-of-dimensionality in kernelized Stein discrepancy (KSD).
method Sliced Stein discrepancy and its scalable variants using optimal one-dimensional projections.
result Significantly outperforms KSD and baselines in goodness-of-fit tests and improves model learning.
This paper solves the inversion problem for jump processes using Markovian projections.
problem Calibrating jump-diffusion models with both local and stochastic features.
method Inverting Markovian projections for pure jump processes.
result Constructs calibrated local stochastic intensity (LSI) models for credit risk applications.
This paper extends Markovian projections to semimartingales with jumps.
problem Extending Markovian projections to semimartingales with jumps.
method Using Markovian projections to match marginal laws of Itô semimartingales with jumps.
result Existence of Markovian projections for Itô semimartingales with jumps.
A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifol…
Sliced Optimal Transport simplifies OT for fast computation.
problem Efficient computation of distances and barycenters for probability measures.
method Combines OT, integral geometry, and statistics for fast computation.
result Retains rich geometric structure while speeding up computations.
We consider complex projective space P^{n} and a smooth closed curve gamma in P^{n}. Harvey and Lawson have defined the notion of the projective hull \hat{K} of a compact subset K in P^n. This concept is an analogue of the polynomial hull of compact subsets of C^{n}. In the present note we study the relation between th…
We prove a global residual formula in terms of logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normal crossing divisors. We also give a formula for the total sum of the logarithmic indices if the singular set of the foliation is contained in the invaria…
In a paper from 1954, Marstrand proved that if K⊂R2 with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we show that if M is a simply connected surface with non-positive curvature, then Marstrand's th…
CCP clusters correlated features and projects them to 1D for efficient dimensionality reduction.
problem Efficiency in handling large datasets with high intrinsic dimensions.
method CCP partitions features into correlated clusters and projects them to 1D based on sample correlations.
result CCP achieves efficient dimensionality reduction without matrix diagonalization.
We extend the Deep Image Prior (DIP) framework to one-dimensional signals. DIP is using a randomly initialized convolutional neural network (CNN) to solve linear inverse problems by optimizing over weights to fit the observed measurements. Our main finding is that properly tuned one-dimensional convolutional architectu…
A new method slices and sums radial kernels faster.
problem Fast computation of large kernel sums in kernel methods.
method Random projections to 1D subspaces and QMC for selecting projections.
result QMC-slicing outperforms existing methods on test datasets.
A new metric for comparing measures on tree systems reduces computational burden.
problem Heavy computation in Optimal Transport problems.
method Introducing tree systems and a novel metric (Tree-Sliced Wasserstein distance on Systems of Lines, TSW-SL).
result TSW-SL performs favorably compared to Sliced Wasserstein and its variants.
VPNet uses variable projection for efficient neural network training.
problem Efficient and interpretable neural network training for signal processing.
method Variable projection (VP) applied to neural networks.
result VPNet achieves fast learning and good accuracy with low computational cost.
We propose fast approximations for the generalized sliced-Wasserstein distance.
problem Efficient approximation of the generalized sliced-Wasserstein distance in high dimensions.
method Deterministic approximations using random projections and concentration of measure results.
result One-dimensional projections of high-dimensional random vectors are approximately Gaussian.
Benguria and Loss have conjectured that, amongst all smooth closed curves of length 2π in the plane, the lowest possible eigenvalue of the operator L=−Δ+κ2 was one. They observed that this value was achieved on a two-parameter family, O, of geometrically distinct ovals containing the round circle and c…
Proposes variance reduction techniques for sliced Wasserstein distance estimation.
problem Intractability of estimating sliced Wasserstein distances.
method Uses control variates based on Gaussian approximations of projected measures.
result Significant reduction in variance of SW distance estimators.
A new method for comparing image probability measures using convolution operators.
problem Efficiently comparing images using conventional sliced Wasserstein methods.
method Proposed convolution sliced Wasserstein (CSW) methods with stride, dilation, and non-linear activation.
result CSW demonstrates favorable performance over conventional sliced Wasserstein in image comparison and deep generative modeling.
Paper proposes PPMM for fast estimation of large-scale OTM.
problem Estimation of large-scale optimal transport maps (OTM) is challenging due to the curse of dimensionality.
method Combines projection pursuit regression and sufficient dimension reduction to adaptively select projection directions.
result PPMM consistently estimates the most informative projection direction and weakly converges to the target OTM.
Regularization and normalization have become indispensable components in training deep neural networks, resulting in faster training and improved generalization performance. We propose the projected error function regularization loss (PER) that encourages activations to follow the standard normal distribution. PER rand…
Study surface subgroups acting on projective space, finding bending laminations and spheres.
problem Surface subgroups acting on RP3 with coaffine representations. method Stratification of convex core boundary, bending laminations, and analysis of holonomy.
result Projectivization of bending data space is a sphere of dimension 6g−7. Gaussian processes (GPs) provide flexible distributions over functions, with inductive biases controlled by a kernel. However, in many applications Gaussian processes can struggle with even moderate input dimensionality. Learning a low dimensional projection can help alleviate this curse of dimensionality, but introduc…
Paper improves ℓ0-SSC for noisy data by proving SDP and proposing Noisy-DR-ℓ0-SSC.
problem Noisy data and less restrictive subspace affinity in sparse subspace clustering.
method Proposes Noisy-DR-ℓ0-SSC, which projects data onto a lower dimensional space and then applies noisy ℓ0-SSC. result Theoretical guarantee on the correctness of noisy ℓ0-SSC in terms of SDP on noisy data. A new method approximates the Sliced-Wasserstein distance without random projections.
problem Efficiently approximating the Sliced-Wasserstein distance for machine learning applications.
method Utilizing the concentration of measure phenomenon to develop a deterministic approximation.
result The approximation error goes to zero as the dimension increases, under a weak dependence condition.
We study multi-moment maps induced by a two-torus action on the four homogeneous nearly Kähler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an SU(3)-structure admi…
The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
Study one-dimensional topological theories with linear generating functions.
problem Understanding one-dimensional topological theories with defects.
method Construct bases of hom spaces for decorated unoriented one-dimensional cobordisms.
result Gram determinant and linear generating functions constructed.
Subspace clustering is the problem of partitioning unlabeled data points into a number of clusters so that data points within one cluster lie approximately on a low-dimensional linear subspace. In many practical scenarios, the dimensionality of data points to be clustered are compressed due to constraints of measuremen…
We study two-dimensional stochastic differential equations (SDEs) of McKean--Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such SDEs arise when one tries to invert the Markovian projection developed …
Researchers find Busemann functions in Wasserstein space, enabling efficient projections and distances.
problem Defining Busemann functions in Wasserstein space for efficient data projections and distances.
method Investigated existence and computation of Busemann functions in Wasserstein space, establishing closed-form expressions for specific cases.
result Explicit projection schemes for probability distributions on \(\mathbb{R}\) enable novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets.
A new model for forward curves captures behavior through a single equation.
problem Modeling forward curves in a complex function space.
method Developed a stochastic partial differential equation with locally state-dependent coefficients.
result The model retains simplicity while capturing entire forward curve behavior.
A new approach simplifies Sliced-Wasserstein distances to improve learning performance.
problem The concentration of measure phenomenon makes random projections uninformative in high dimensions.
method Propose rescaling the 1D Wasserstein distance to make all slices equally informative.
result The classical Sliced-Wasserstein, properly configured, can match or surpass complex variants.
New neural networks learn single-index models efficiently.
problem Learning low-dimensional structure in high-dimensional data.
method Shallow neural networks with frozen biases, studied via gradient flow.
result Generalization guarantees match near-optimal sample complexity.
New algorithm reduces regret for single-index bandits to nearly optimal.
problem Optimizing rewards from unknown projections of high-dimensional contexts.
method Two-phase algorithm: estimate projection direction, reduce to 1D bandit, use UCB.
result Achieved optimal regret of ildeO(T2/3). Improved regret for single-index bandits with optimal algorithm.
problem Optimizing rewards from unknown one-dimensional projections of high-dimensional contexts.
method Two-phase algorithm: first estimate projection direction, then discretize and use UCB.
result Achieved optimal regret of ildeO(T2/3).