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Quantifying dependence between high-dimensional random variables is central to statistical learning and inference. Two classical methods are canonical correlation analysis (CCA), which identifies maximally correlated projected versions of the original variables, and Shannon's mutual information, which is a universal dependence measure that also captures high-order dependencies. However, CCA only accounts for linear dependence, which may be insufficient for certain applications, while mutual information is often infeasible to compute/estimate in high dimensions. This work proposes a middle ground in the form of a scalable information-theoretic generalization of CCA, termed max-sliced mutual information (mSMI). mSMI equals the maximal mutual information between low-dimensional projections of the high-dimensional variables, which reduces back to CCA in the Gaussian case. It enjoys the best of both worlds: capturing intricate dependencies in the data while being amenable to fast computation and scalable estimation from samples. We show that mSMI retains favorable structural properties of Shannon's mutual information, like variational forms and identification of independence. We then study statistical estimation of mSMI, propose an efficiently computable neural estimator, and couple it with formal non-asymptotic error bounds. We present experiments that demonstrate the utility of mSMI for several tasks, encompassing independence testing, multi-view representation learning, algorithmic fairness, and generative modeling. We observe that mSMI consistently outperforms competing methods with little-to-no computational overhead.
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Marginal Distance and Hilbert-Schmidt Covariances-Based Independence Tests for Multivariate Functional Data
We study the pairwise and mutual independence testing problem for multivariate functional data. Using a basis representation of functional data, we reduce this problem to testing the independence of multivariate data, which may be high-dimensional. For pairwise independence, we apply tests based on distance and Hilbert-Schmidt covariances as well as their marginal versions, which aggregate these covariances for coordinates of random processes. In the case of mutual independence, we study asymmetric and symmetric aggregating measures of pairwise dependence. A theoretical justification of the test procedures is established. In extensive simulation studies and examples based on a real economic data set, we investigate and compare the performance of the tests in terms of size control and power. An important finding is that tests based on distance and Hilbert-Schmidt covariances are usually more powerful than their marginal versions under linear dependence, while the reverse is true under non-linear dependence.
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- PAR ID:
- 10332107
- Date Published:
- Journal Name:
- Journal of Artificial Intelligence Research
- Volume:
- 73
- ISSN:
- 1076-9757
- Page Range / eLocation ID:
- 1355 to 1384
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1613/jair.1.13233
