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Consider the setting of Linear Regression where the observed response variables, in expectation, are linear functions of the pdimensional covariates. Then to achieve vanishing prediction error, the number of required samples scales faster than pσ2, where σ2 is a bound on the noise variance. In a highdimensional setting where p is large but the covariates admit a lowdimensional representation (say r ≪ p), then Principal Component Regression (PCR), cf. [36], is an effective approach; here, the response variables are regressed with respect to the principal components of the covariates. The resulting number of required samples to achieve vanishing prediction error now scales faster than rσ2(≪ pσ2). Despite the tremendous utility of PCR, its ability to handle settings with noisy, missing, and mixed (discrete and continuous) valued covariates is not understood and remains an important open challenge, cf. [24]. As the main contribution of this work, we address this challenge by rigorously establishing that PCR is robust to noisy, sparse, and possibly mixed valued covariates. Specifically, under PCR, vanishing prediction error is achieved with the number of samples scaling as r max(σ2, ρ−4 log5(p)), where ρ denotes the fraction of observed (noisy) covariates. We establish generalization error bounds on the performance of PCR, which provides a systematic approach in selecting the correct number of components r in a datadriven manner. The key to our result is a simple, but powerful equivalence between (i) PCR and (ii) Linear Regression with covariate preprocessing via Hard Singular Value Thresholding (HSVT). From a technical standpoint, this work advances the stateoftheart analysis for HSVT by establishing stronger guarantees with respect to the ∥·∥2,∞error for the estimated matrix rather than the Frobenius norm/meansquared error (MSE) as is commonly done in the matrix estimation / completion literature.
Author Information
Anish Agarwal (MIT)
Devavrat Shah (Massachusetts Institute of Technology)
Devavrat Shah is a professor of Electrical Engineering & Computer Science and Director of Statistics and Data Science at MIT. He received PhD in Computer Science from Stanford. He received Erlang Prize from Applied Probability Society of INFORMS in 2010 and NeuIPS best paper award in 2008.
Dennis Shen (Massachusetts Institute of Technology)
Dogyoon Song (Massachusetts Institute of Technology)
Related Events (a corresponding poster, oral, or spotlight)

2019 Oral: On Robustness of Principal Component Regression »
Fri Dec 13th 12:25  12:40 AM Room West Exhibition Hall C + B3
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