Randomised regression

Tackling your regression, by using random projections of the predictors.

Usually this means using those projections to reduce the dimensionality of a high dimensional regression. In this case it is not far from compressed sensing, except in how we handle noise. In this linear model case, this is of course random linear algebra, and may be a randomised matrix factorisation.

Occasionally we might use non-linear projections to increase the dimensionality of our data in the hope of making a non-linear regression approximately linear, which dates back to (Cover 1965).

I am especially interested in seeing how this might be useful for dependent data, especially time series.

Brian McWilliams, Gabriel Krummenacher and Mario Lučić, Randomized Linear Regression: A brief overview and recent results. Gabriel implemented some of the algorithms mentioned, e.g.

  • Subsampled Randomized Fourier Transform.

  • SRHT: Uses the Subsampled Randomized Hadamard Transform (SRHT), equivalent to leverage based sampling.

  • aIWS, aRWS: Samples based on approximated statistical influence.

  • Uluru: SRHT with bias correction.

Martin Wainright, Statistics meets Optimization: Randomization and approximation for high-dimensional problems.

In the modern era of high-dimensional data, the interface between mathematical statistics and optimization has become an increasingly vibrant area of research. In this course, we provide some vignettes into this interface, including the following topics:

  • Dimensionality reduction via random projection. The naive idea of projecting high-dimensional data to a randomly chosen low-dimensional space is remarkably effective. We discuss the classical Johnson-Lindenstrauss lemma, as well as various modern variants that provide computationally-efficient embeddings with strong guarantees.

  • When is it possible to quickly obtain approximate solutions of large-scale convex programs? In practice, methods based on randomized projection can work very well, and arguments based on convex analysis and concentration of measure provide a rigorous underpinning to these observations.

  • Optimization problems with some form of nonconvexity arise frequently in statistical settings -- for instance, in problems with latent variables, combinatorial constraints, or rank constraints. Nonconvex programs are known to be intractable in a complexity-theoretic sense, but the random ensembles arising in statistics are not adversarially constructed. Under what conditions is it possible to make rigorous guarantees about the behavior of simple iterative algorithms for such problems? We develop some general theory for addressing these questions, exploiting tools from both optimization theory and empirical process theory.

Bahmani, Sohail, and Justin Romberg. 2017. “Anchored Regression: Solving Random Convex Equations via Convex Programming,” February. http://arxiv.org/abs/1702.05327.

Choromanski, Krzysztof, Mark Rowland, and Adrian Weller. 2017. “The Unreasonable Effectiveness of Random Orthogonal Embeddings,” March. http://arxiv.org/abs/1703.00864.

Cover, T. M. 1965. “Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition.” IEEE Transactions on Electronic Computers EC-14 (3): 326–34. https://doi.org/10.1109/PGEC.1965.264137.

Dhillon, Paramveer, Yichao Lu, Dean P. Foster, and Lyle Ungar. 2013. “New Subsampling Algorithms for Fast Least Squares Regression.” In Advances in Neural Information Processing Systems, 360–68. Curran Associates, Inc. http://papers.nips.cc/paper/5105-new-subsampling-algorithms-for-fast-least-squares-regression.

Gilbert, Anna C., Yi Zhang, Kibok Lee, Yuting Zhang, and Honglak Lee. 2017. “Towards Understanding the Invertibility of Convolutional Neural Networks,” May. http://arxiv.org/abs/1705.08664.

Gottwald, Georg A., and Sebastian Reich. 2020. “Supervised Learning from Noisy Observations: Combining Machine-Learning Techniques with Data Assimilation,” July. http://arxiv.org/abs/2007.07383.

Gribonval, Rémi, Gilles Blanchard, Nicolas Keriven, and Yann Traonmilin. 2017. “Compressive Statistical Learning with Random Feature Moments,” June. http://arxiv.org/abs/1706.07180.

Gupta, Pawan, and Marianna Pensky. 2016. “Solution of Linear Ill-Posed Problems Using Random Dictionaries,” May. http://arxiv.org/abs/1605.07913.

Heinze, Christina, Brian McWilliams, and Nicolai Meinshausen. 2016. “DUAL-LOCO: Distributing Statistical Estimation Using Random Projections.” In, 875–83. http://www.jmlr.org/proceedings/papers/v51/heinze16.html.

Heinze, Christina, Brian McWilliams, Nicolai Meinshausen, and Gabriel Krummenacher. 2014. “LOCO: Distributing Ridge Regression with Random Projections,” June. http://arxiv.org/abs/1406.3469.

Krummenacher, Gabriel, Brian McWilliams, Yannic Kilcher, Joachim M Buhmann, and Nicolai Meinshausen. 2016. “Scalable Adaptive Stochastic Optimization Using Random Projections.” In Advances in Neural Information Processing Systems 29, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, 1750–8. Curran Associates, Inc. http://papers.nips.cc/paper/6054-scalable-adaptive-stochastic-optimization-using-random-projections.pdf.

McWilliams, Brian, David Balduzzi, and Joachim M Buhmann. 2013. “Correlated Random Features for Fast Semi-Supervised Learning.” In Advances in Neural Information Processing Systems 26, edited by C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, 1050:440–48. Curran Associates, Inc. http://papers.nips.cc/paper/5000-correlated-random-features-for-fast-semi-supervised-learning.pdf.

McWilliams, Brian, Gabriel Krummenacher, Mario Lucic, and Joachim M. Buhmann. 2014. “Fast and Robust Least Squares Estimation in Corrupted Linear Models.” In Advances in Neural Information Processing Systems, 415–23. http://papers.nips.cc/paper/5428-fast-and-robust-least-squares-estimation-in-corrupted-linear-models.

Rosenfeld, Amir, and John K. Tsotsos. 2018. “Intriguing Properties of Randomly Weighted Networks: Generalizing While Learning Next to Nothing.”

Scardapane, Simone, and Dianhui Wang. 2017. “Randomness in Neural Networks: An Overview.” Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 7 (2). https://doi.org/10.1002/widm.1200.

Soni, Akshay, and Yashar Mehdad. 2017. “RIPML: A Restricted Isometry Property Based Approach to Multilabel Learning,” February. http://arxiv.org/abs/1702.05181.

Thanei, Gian-Andrea, Christina Heinze, and Nicolai Meinshausen. 2017. “Random Projections for Large-Scale Regression,” January. http://arxiv.org/abs/1701.05325.

Wang, HaiYing, Rong Zhu, and Ping Ma. 2017. “Optimal Subsampling for Large Sample Logistic Regression,” February. http://arxiv.org/abs/1702.01166.

Zhang, Xiao, Lingxiao Wang, and Quanquan Gu. 2017. “Stochastic Variance-Reduced Gradient Descent for Low-Rank Matrix Recovery from Linear Measurements,” January. http://arxiv.org/abs/1701.00481.