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research
Optimal linear estimation under unknown nonlinear transform
Authors
Constantine Caramanis
Han Liu
Zhaoran Wang
Xinyang Yi
Publication date
1 January 2015
Publisher
View
on
arXiv
Abstract
Linear regression studies the problem of estimating a model parameter
Ξ²
β
β
R
p
\beta^* \in \mathbb{R}^p
Ξ²
β
β
R
p
, from
n
n
n
observations
{
(
y
i
,
x
i
)
}
i
=
1
n
\{(y_i,\mathbf{x}_i)\}_{i=1}^n
{(
y
i
β
,
x
i
β
)
}
i
=
1
n
β
from linear model
y
i
=
β¨
x
i
,
Ξ²
β
β©
+
Ο΅
i
y_i = \langle \mathbf{x}_i,\beta^* \rangle + \epsilon_i
y
i
β
=
β¨
x
i
β
,
Ξ²
β
β©
+
Ο΅
i
β
. We consider a significant generalization in which the relationship between
β¨
x
i
,
Ξ²
β
β©
\langle \mathbf{x}_i,\beta^* \rangle
β¨
x
i
β
,
Ξ²
β
β©
and
y
i
y_i
y
i
β
is noisy, quantized to a single bit, potentially nonlinear, noninvertible, as well as unknown. This model is known as the single-index model in statistics, and, among other things, it represents a significant generalization of one-bit compressed sensing. We propose a novel spectral-based estimation procedure and show that we can recover
Ξ²
β
\beta^*
Ξ²
β
in settings (i.e., classes of link function
f
f
f
) where previous algorithms fail. In general, our algorithm requires only very mild restrictions on the (unknown) functional relationship between
y
i
y_i
y
i
β
and
β¨
x
i
,
Ξ²
β
β©
\langle \mathbf{x}_i,\beta^* \rangle
β¨
x
i
β
,
Ξ²
β
β©
. We also consider the high dimensional setting where
Ξ²
β
\beta^*
Ξ²
β
is sparse ,and introduce a two-stage nonconvex framework that addresses estimation challenges in high dimensional regimes where
p
β«
n
p \gg n
p
β«
n
. For a broad class of link functions between
β¨
x
i
,
Ξ²
β
β©
\langle \mathbf{x}_i,\beta^* \rangle
β¨
x
i
β
,
Ξ²
β
β©
and
y
i
y_i
y
i
β
, we establish minimax lower bounds that demonstrate the optimality of our estimators in both the classical and high dimensional regimes.Comment: 25 pages, 3 figure
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Last time updated on 14/02/2024