On matrix estimation under monotonicity constraints
| Autor: | Adityanand Guntuboyina, Sabyasachi Chatterjee, Bodhisattva Sen |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
Statistics::Theory Logarithm Monotonic function Mathematics - Statistics Theory 02 engineering and technology Statistics Theory (math.ST) adaptation 01 natural sciences variable adaptation Combinatorics 010104 statistics & probability Matrix (mathematics) Computer Science::Systems and Control 0202 electrical engineering electronic engineering information engineering FOS: Mathematics oracle inequalities 0101 mathematics Mathematics minimax lower bound tangent cone Convex geometry Multiplicative function 020206 networking & telecommunications Minimax Bounded function metric entropy bounds bivariate isotonic regression Constant (mathematics) |
| Zdroj: | Bernoulli 24, no. 2 (2018), 1072-1100 |
| Popis: | We consider the problem of estimating an unknown $n_{1}\times n_{2}$ matrix $\mathbf{\theta}^{*}$ from noisy observations under the constraint that $\mathbf{\theta}^{*}$ is nondecreasing in both rows and columns. We consider the least squares estimator (LSE) in this setting and study its risk properties. We show that the worst case risk of the LSE is $n^{-1/2}$, up to multiplicative logarithmic factors, where $n=n_{1}n_{2}$ and that the LSE is minimax rate optimal (up to logarithmic factors). We further prove that for some special $\mathbf{\theta}^{*}$, the risk of the LSE could be much smaller than $n^{-1/2}$; in fact, it could even be parametric, that is, $n^{-1}$ up to logarithmic factors. Such parametric rates occur when the number of “rectangular” blocks of $\mathbf{\theta}^{*}$ is bounded from above by a constant. We also derive an interesting adaptation property of the LSE which we term variable adaptation – the LSE adapts to the “intrinsic dimension” of the problem and performs as well as the oracle estimator when estimating a matrix that is constant along each row/column. Our proofs, which borrow ideas from empirical process theory, approximation theory and convex geometry, are of independent interest. |
| Databáze: | OpenAIRE |
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