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March Boedihardjo

Publications and source records attributed to March Boedihardjo.

15 recordsLinked to original sources
arXiv2024

Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances

Optimal transport has been very successful for various machine learning tasks; however, it is known to suffer from the curse of dimensionality. Hence, dimensionality reduction is desirable when applied to high-dimensional data with low-dimensional structures. The kernel max-sliced (KMS) Wasserstein distance is developed for this purpose by finding an optimal nonlinear mapping that reduces data into $1$ dimension before computing the Wasserstein distance. However, its theoretical properties have not yet been fully developed. In this paper, we provide sharp finite-sample guarantees under milder technical assumptions compared with state-of-the-art for the KMS $p$-Wasserstein distance between two empirical distributions with $n$ samples for general $p\in[1,\infty)$. Algorithm-wise, we show that computing the KMS $2$-Wasserstein distance is NP-hard, and then we further propose a semidefinite relaxation (SDR) formulation (which can be solved efficiently in polynomial time) and provide a relaxation gap for the obtained solution. We provide numerical examples to demonstrate the good performance of our scheme for high-dimensional two-sample testing.

stat.ML↗
arXiv2024

Metric geometry of the privacy-utility tradeoff

Synthetic data are an attractive concept to enable privacy in data sharing. A fundamental question is how similar the privacy-preserving synthetic data are compared to the true data. Using metric privacy, an effective generalization of differential privacy beyond the discrete setting, we raise the problem of characterizing the optimal privacy-accuracy tradeoff by the metric geometry of the underlying space. We provide a partial solution to this problem in terms of the "entropic scale", a quantity that captures the multiscale geometry of a metric space via the behavior of its packing numbers. We illustrate the applicability of our privacy-accuracy tradeoff framework via a diverse set of examples of metric spaces.

cs.CR↗
arXiv2024

Private measures, random walks, and synthetic data

Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex - but very common - machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data for general compact metric spaces. A key ingredient in our construction is a new superregular random walk, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmicaly slowly.

cs.CR↗
arXiv2023

Certified private data release for sparse Lipschitz functions

As machine learning has become more relevant for everyday applications, a natural requirement is the protection of the privacy of the training data. When the relevant learning questions are unknown in advance, or hyper-parameter tuning plays a central role, one solution is to release a differentially private synthetic data set that leads to similar conclusions as the original training data. In this work, we introduce an algorithm that enjoys fast rates for the utility loss for sparse Lipschitz queries. Furthermore, we show how to obtain a certificate for the utility loss for a large class of algorithms.

cs.CR↗
arXiv2023

On algebra-valued R-diagonal elements (with erratum)

For an element in an algebra-valued *-noncommutative probability space, equivalent conditions for algebra-valued R-diagonality (a notion introduced by Sniady and Speicher) are proved. Formal power series relations involving the moments and cumulants of such R-diagonal elements are proved. Decompositions of algebra-valued R-diagonal elements into products of the form unitary times self-adjoint are investigated; sufficient conditions, in terms of cumulants, for *-freeness of the unitary and the self-adjoint part are proved, and a tracial example is given where *-freeness fails. The particular case of algebra-valued circular elements is considered; as an application, the polar decompostion of the quasinilpotent DT-operator is described. An erratum is appended. (The last sentence in the above paragraph is thereby nullified.)

math.OA↗
arXiv2023

Covariance loss, Szemeredi regularity, and differential privacy

We show how randomized rounding based on Grothendieck's identity can be used to prove a nearly tight bound on the covariance loss--the amount of covariance that is lost by taking conditional expectation. This result yields a new type of weak Szemeredi regularity lemma for positive semidefinite matrices and kernels. Moreover, it can be used to construct differentially private synthetic data.

math.PR↗
arXiv2022

Covariance's Loss is Privacy's Gain: Computationally Efficient, Private and Accurate Synthetic Data

The protection of private information is of vital importance in data-driven research, business, and government. The conflict between privacy and utility has triggered intensive research in the computer science and statistics communities, who have developed a variety of methods for privacy-preserving data release. Among the main concepts that have emerged are anonymity and differential privacy. Today, another solution is gaining traction, synthetic data. However, the road to privacy is paved with NP-hard problems. In this paper we focus on the NP-hard challenge to develop a synthetic data generation method that is computationally efficient, comes with provable privacy guarantees, and rigorously quantifies data utility. We solve a relaxed version of this problem by studying a fundamental, but a first glance completely unrelated, problem in probability concerning the concept of covariance loss. Namely, we find a nearly optimal and constructive answer to the question how much information is lost when we take conditional expectation. Surprisingly, this excursion into theoretical probability produces mathematical techniques that allow us to derive constructive, approximately optimal solutions to difficult applied problems concerning microaggregation, privacy, and synthetic data.

cs.CR↗
arXiv2021

Private sampling: a noiseless approach for generating differentially private synthetic data

In a world where artificial intelligence and data science become omnipresent, data sharing is increasingly locking horns with data-privacy concerns. Differential privacy has emerged as a rigorous framework for protecting individual privacy in a statistical database, while releasing useful statistical information about the database. The standard way to implement differential privacy is to inject a sufficient amount of noise into the data. However, in addition to other limitations of differential privacy, this process of adding noise will affect data accuracy and utility. Another approach to enable privacy in data sharing is based on the concept of synthetic data. The goal of synthetic data is to create an as-realistic-as-possible dataset, one that not only maintains the nuances of the original data, but does so without risk of exposing sensitive information. The combination of differential privacy with synthetic data has been suggested as a best-of-both-worlds solutions. In this work, we propose the first noisefree method to construct differentially private synthetic data; we do this through a mechanism called "private sampling". Using the Boolean cube as benchmark data model, we derive explicit bounds on accuracy and privacy of the constructed synthetic data. The key mathematical tools are hypercontractivity, duality, and empirical processes. A core ingredient of our private sampling mechanism is a rigorous "marginal correction" method, which has the remarkable property that importance reweighting can be utilized to exactly match the marginals of the sample to the marginals of the population.

cs.CR↗
arXiv2021

Privacy of synthetic data: a statistical framework

Privacy-preserving data analysis is emerging as a challenging problem with far-reaching impact. In particular, synthetic data are a promising concept toward solving the aporetic conflict between data privacy and data sharing. Yet, it is known that accurately generating private, synthetic data of certain kinds is NP-hard. We develop a statistical framework for differentially private synthetic data, which enables us to circumvent the computational hardness of the problem. We consider the true data as a random sample drawn from a population Omega according to some unknown density. We then replace Omega by a much smaller random subset Omega^*, which we sample according to some known density. We generate synthetic data on the reduced space Omega^* by fitting the specified linear statistics obtained from the true data. To ensure privacy we use the common Laplacian mechanism. Employing the concept of Renyi condition number, which measures how well the sampling distribution is correlated with the population distribution, we derive explicit bounds on the privacy and accuracy provided by the proposed method.

cs.CR↗
arXiv2020

A Performance Guarantee for Spectral Clustering

The two-step spectral clustering method, which consists of the Laplacian eigenmap and a rounding step, is a widely used method for graph partitioning. It can be seen as a natural relaxation to the NP-hard minimum ratio cut problem. In this paper we study the central question: when is spectral clustering able to find the global solution to the minimum ratio cut problem? First we provide a condition that naturally depends on the intra- and inter-cluster connectivities of a given partition under which we may certify that this partition is the solution to the minimum ratio cut problem. Then we develop a deterministic two-to-infinity norm perturbation bound for the the invariant subspace of the graph Laplacian that corresponds to the $k$ smallest eigenvalues. Finally by combining these two results we give a condition under which spectral clustering is guaranteed to output the global solution to the minimum ratio cut problem, which serves as a performance guarantee for spectral clustering.

stat.ML↗
arXiv2019

DP-LSSGD: A Stochastic Optimization Method to Lift the Utility in Privacy-Preserving ERM

Machine learning (ML) models trained by differentially private stochastic gradient descent (DP-SGD) have much lower utility than the non-private ones. To mitigate this degradation, we propose a DP Laplacian smoothing SGD (DP-LSSGD) to train ML models with differential privacy (DP) guarantees. At the core of DP-LSSGD is the Laplacian smoothing, which smooths out the Gaussian noise used in the Gaussian mechanism. Under the same amount of noise used in the Gaussian mechanism, DP-LSSGD attains the same DP guarantee, but in practice, DP-LSSGD makes training both convex and nonconvex ML models more stable and enables the trained models to generalize better. The proposed algorithm is simple to implement and the extra computational complexity and memory overhead compared with DP-SGD are negligible. DP-LSSGD is applicable to train a large variety of ML models, including DNNs. The code is available at \url{https://github.com/BaoWangMath/DP-LSSGD}.

cs.LG↗
arXiv2017

Asymptotic *-moments of some random Vandermonde matrices

Appropriately normalized square random Vandermonde matrices based on independent random variables with uniform distribution on the unit circle are studied. It is shown that as the matrix sizes increases without bound, with respect to the expectation of the trace there is an asymptotic *-distribution, equal to that of a C[0,1]-valued R-diagonal element.

math.PR↗
arXiv2017

Multiplication operators on L^p

We show that every operator on $L^{p}$, $1<p<\infty$ defined by multiplication by the identity function on $\mathbb{C}$ is a compact perturbation of an operator that is diagonal with respect to an unconditional basis. We also classify these operators up to similarity modulo compact operators and up to approximate similarity.

math.FA↗
arXiv2015

Maximum moments of sum of independent random matrices

We show that the maximum moments of the sum of independent positive semidefinite random matrices with given norm upper bounds and norms of expectations is attained when all the random matrices are the multiplications of certain random variables and the identity matrix.

math.PR↗