Determinantal averaging corrects inversion bias in distributed Newton's method.
problem Inverting a sum of distributed matrices is biased; local averages are incorrect.
method Reweighting local estimates of the Newton's step proportionally to the determinant of the local Hessian estimate, then averaging them.
result Determinantal averaging provides the first known asymptotically consistent distributed Newton step.
The paper develops efficient algorithms for sampling from random spanning trees and determinantal point processes.
problem Sampling from strongly Rayleigh distributions efficiently.
method Optimal sublinear sampling algorithms for random spanning trees and determinantal point processes.
result Achieves optimal sublinear sampling for strongly Rayleigh distributions.
In this technical report, we discuss several sampling algorithms for Determinantal Point Processes (DPP). DPPs have recently gained a broad interest in the machine learning and statistics literature as random point processes with negative correlation, i.e., ones that can generate a "diverse" sample from a set of items.…
Study characteristic classes of a specific type of determinantal varieties.
problem Understanding the geometric properties of a special class of determinantal varieties.
method Used Schubert calculus to derive explicit formulas for Chern-Schwartz-MacPherson and Chern-Mather classes.
result Explicit formulas for sectional Euler characteristics, characteristic cycles, and polar classes were obtained.
This paper improves signal reconstruction using determinantal sampling from random nodes.
problem Approximating square-integrable functions from random node evaluations.
method Combines determinantal point processes and mixtures thereof for RKHS-adapted approximations.
result Proves mean-square guarantees in L2 norm and shows faster convergence rates. Adaptive sampling for risk-averse learning on hard examples.
problem Training models to perform well on difficult examples in high-stakes applications.
method Adaptive sampling algorithm for stochastically optimizing CVaR, using distributionally robust formulation and regret minimization.
result Empirically demonstrates effectiveness on large-scale convex and non-convex learning tasks.
We study the Euler obstruction of essentially isolated determinantal singularities (EIDS). The EIDS were defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We obtain some formulas to calculate the Euler obstruction for the determinantal varieties with singular set an ICIS.
Study Euler obstruction of 1-forms on determinantal singularities.
problem Understanding the Euler obstruction of 1-forms on determinantal singularities.
method Investigation of connections between local Euler obstruction and PHN index.
result Explicit computations of Euler obstruction for specific singularities.
Minimal cones defined by rank conditions in matrix spaces.
problem Characterizing minimal cones in matrix spaces.
method Proving determinantal varieties are minimal cones with regular strata.
result Determinantal varieties are minimal cones with regular submanifolds.
Determinantal point process have recently been used as models in machine learning and this has raised questions regarding the characterizations of conditional independence. In this paper we investigate characterizations of conditional independence. We describe some conditional independencies through the conditions on t…
We study U(N|M) character expectation value with the supermatrix Chern-Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We deri…
In this note we consider sampling from (non-homogeneous) strongly Rayleigh probability measures. As an important corollary, we obtain a fast mixing Markov Chain sampler for Determinantal Point Processes.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to projective varieties with isolated singularities.
method Using generalized Poincaré-Hopf indices for a projective variety with isolated determinantal singularities.
result A Poincaré-Hopf type theorem is proven for projective varieties with isolated singularities.
This paper proves all Pfaffian varieties are area-minimizing except hypersurfaces.
problem Proving area-minimizing property of Pfaffian varieties.
method Analyzing families of minimal real matrix varieties and proving area-minimizing property.
result All Pfaffian varieties are area-minimizing except hypersurfaces.
Paper proposes a simple estimator for DPP correlation kernels.
problem Estimating the correlation kernel matrix of DPPs.
method Closed-form estimator for correlation kernel, easy to implement.
result Consistency and asymptotic normality of the estimator proved.
Given a fixed n×d matrix X, where n≫d, we study the complexity of sampling from a distribution over all subsets of rows where the probability of a subset is proportional to the squared volume of the parallelepiped spanned by the rows (a.k.a. a determinantal point process). In this task, it is im…
We calculate the free energy of Coulomb gas systems on Riemann surfaces.
problem Analyzing the free energy of Coulomb gas systems on Riemann surfaces.
method Using bosonization formula and analytic torsion, we derive the asymptotic expansion of the partition function.
result We prove the geometric version of the Zabrodin-Wiegmann conjecture in the determinantal case.
Paper explores duality in DPPs using embedding structure analysis.
problem Understanding the geometric structure of determinantal point processes.
method Analyzes the exponential family embedding of DPPs and uses the e-embedding curvature tensor.
result Discovers the duality between marginal and L-ensemble kernels.
Determinantal consensus clustering improves clustering robustness.
problem Robustness of clustering algorithms.
method Use of determinantal point processes (DPP) for random restart of clustering algorithms.
result Determinantal consensus clustering outperforms classical algorithms.
New algorithm samples determinantal point processes with sublinear preprocessing time.
problem Sampling from determinantal point processes efficiently with small expected subset size.
method Proposes an algorithm with sublinear preprocessing and independent sampling cost.
result Achieves sublinear preprocessing time and independent sampling cost.
The study examines determinantal point processes linked to a specific operator on Riemannian manifolds.
problem Understanding the spectral properties and associated point processes of the Bochner-Schrödinger operator.
method Analysis of the Bochner-Schrödinger operator on tensor powers of Hermitian line bundles, focusing on large p asymptotics. result The asymptotic behavior of determinantal point processes associated with the operator's spectral projection is computed, leading to the law of large numbers and central limit theorem.
This research uses DPPs to improve semi-parametric regression models.
problem Improving comprehensibility in semi-parametric regression models without sacrificing accuracy.
method Introduced a novel representation of finite DPPs and used it to derive a key identity illustrating implicit regularization.
result Demonstrated the implicit regularization effect of determinantal sampling for semi-parametric regression.
The study computes Bergman kernels and point process asymptotics on Kähler manifolds.
problem Computing asymptotics of Bergman kernels and point process distributions on Kähler manifolds.
method Equivariant and partial Bergman kernels, determinantal point processes, asymptotic analysis.
result The distribution of linear statistics converges to a centered normal variable with specific variances.
This work improves sampling efficiency on complex spaces using determinantal processes.
problem Efficient sampling from large-scale datasets with general spaces.
method Determinantal point processes on general spaces and diffusion geometry.
result Improved sampling rates for determinantal processes on Riemannian manifolds and networks.
New algorithms for online MAP inference and learning for NDPPs.
problem Online inference and learning for nonsymmetric determinantal point processes.
method Single-pass algorithms with sub-linear memory usage.
result Comparable performance to offline algorithms with multiple passes.
The paper studies partition functions of point processes on Kähler manifolds, generalizing geometric functionals and relating to QHE.
problem Analyzing partition functions of determinantal point processes on Kähler manifolds.
method Using geometric functionals and TYZ expansion coefficients of the Bergman kernel.
result The coefficients of the partition function expansion are geometric functionals on Kähler metrics.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to varieties with isolated singularities.
method Using generalizations of the Poincaré-Hopf index.
result A Poincaré-Hopf type theorem for projective varieties with isolated singularities.
New sampling method reduces variance in correlated high-dimensional distributions.
problem Reducing variance in Monte Carlo estimators for correlated high-dimensional distributions.
method DPPMC (Determinantal Point Processes Monte Carlo) method for structured sampling.
result DPPMCs improve state-of-the-art in various optimization and machine learning problems.
The paper finds determinantal expressions for certain symmetric space integrals.
problem Finding compact expressions for integrals on symmetric spaces.
method Expressing integrals as determinants or Pfaffians for K-invariant functions. result Determinantal expressions for specific symmetric space integrals.
New algorithm scales NDPP learning and inference to large item collections.
problem Memory and runtime limitations in existing NDPP learning and inference algorithms.
method Introduced a new NDPP kernel decomposition for learning and a linear-complexity MAP inference algorithm.
result Our algorithms scale linearly in M, matching prior work's predictive performance. Determinantal point processes (DPPs) are probabilistic models for repulsion. When used to represent the occurrence of random subsets of a finite base set, DPPs allow to model global negative associations in a mathematically elegant and direct way. Discrete DPPs have become popular and computationally tractable models f…
We propose a new class of determinantal point processes (DPPs) which can be manipulated for inference and parameter learning in potentially sublinear time in the number of items. This class, based on a specific low-rank factorization of the marginal kernel, is particularly suited to a subclass of continuous DPPs and DP…
Paper explores how DPP sampling can implicitly regularize kernel regression.
problem Improving kernel regression by reducing redundancy in data.
method Using Determinantal Point Processes (DPPs) to sample subsets implicitly regularizes ridgeless Kernel Regression.
result Ensemble of ridgeless regressors can be effective for datasets with redundant information.
Faster sampler reduces DPP sampling cost to O(nm + m^3 log m).
problem High cost of sampling discrete DPPs.
method Uses rejection sampling and leverage score i.i.d. sampling.
result Reduces sampling cost from O(n^3) to O(nm + m^3 log m).
DPP-BBO diversifies batched Bayesian optimization using DPPs.
problem Efficiently proposing diverse and informative batches in batched Bayesian optimization.
method Introducing DPP-Batch Bayesian Optimization (DPP-BBO) with DPP-Thompson Sampling (DPP-TS).
result Novel Bayesian simple regret bounds for DPP-TS show improved performance over classical methods.
Determinantal point processes (DPPs) are well-suited for modeling repulsion and have proven useful in many applications where diversity is desired. While DPPs have many appealing properties, such as efficient sampling, learning the parameters of a DPP is still considered a difficult problem due to the non-convex nature…
The paper examines when real matrix Schubert varieties are minimal submanifolds.
problem When are real matrix Schubert varieties minimal submanifolds?
method The authors establish minimality conditions using geometric arguments and partial permutations.
result The paper identifies specific conditions for real matrix Schubert varieties to be minimal submanifolds.
If E is a C^\infty complex vector bundle on an oriented C^\infty manifold Σ, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal gerbe with lien C^*, the group of nonzero complex numbers. If q:Σ-->B is a smooth famil…
Determinantal point processes (DPPs) are elegant probabilistic models of repulsion that arise in quantum physics and random matrix theory. In contrast to traditional structured models like Markov random fields, which become intractable and hard to approximate in the presence of negative correlations, DPPs offer efficie…
This work improves SGD minibatch sampling using determinantal point processes based on orthogonal polynomials.
problem Improving variance reduction in stochastic gradient descent (SGD) for large datasets.
method Orthogonal polynomial-based determinantal point processes for sampling minibatches in SGD.
result DPP minibatches lead to a smaller mean square approximation error than uniform minibatches.
We present a new random sampling strategy for k-bandlimited signals defined on graphs, based on determinantal point processes (DPP). For small graphs, ie, in cases where the spectrum of the graph is accessible, we exhibit a DPP sampling scheme that enables perfect recovery of bandlimited signals. For large graphs, ie, …
Paper tests DPPs for diversity models, distinguishing them from other distributions.
problem Testing whether a given distribution is a Determinantal Point Process (DPP) or far from any DPP.
method Proposes the first algorithm for DPP testing and establishes a lower bound on sample complexity.
result Establishes a matching lower bound on the sample complexity of DPP testing.
Determinantal Point Processes (DPPs) are probabilistic models over all subsets a ground set of N items. They have recently gained prominence in several applications that rely on "diverse" subsets. However, their applicability to large problems is still limited due to the O(N3) complexity of core tasks suc…
Determinantal point processes (DPPs) have received significant attention in the recent years as an elegant model for a variety of machine learning tasks, due to their ability to elegantly model set diversity and item quality or popularity. Recent work has shown that DPPs can be effective models for product recommendati…
A new point process for clustering distributions with repulsion.
problem Clustering distributions with repulsion.
method Distributional Determinantal Point Process (dDPP) with sliced Wasserstein kernel.
result Validated dDPP as a well-defined point process and applied to gene expression and epilepsy data.
Study the limits of discrete DPPs to continuous DPPs as set size grows.
problem Characterize the behavior of discrete DPPs as they approach continuous DPPs.
method Non-asymptotic characterization of the limit in terms of weak coherency.
result Sufficient conditions for weak coherency are identified.
Quantum machine learning boosts financial forecasting accuracy.
problem Churn prediction and credit risk assessment in finance.
method Used quantum and classical Determinantal Point Processes for churn prediction, and quantum neural networks for credit risk assessment.
result Significant improvement in precision for churn prediction (6% increase). Quantum models match classical performance with fewer parameters.
New algorithms improve experimental design efficiency and approximation quality.
problem Finding optimal subset of vectors for expensive measurements.
method Bayesian experimental design using determinantal point processes.
result Developed efficient algorithms for optimal design under multiple criteria.