Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Performance bound of the intensity-based model for noisy phase retrieval   The aim of noisy phase retrieval is to estimate a signal $\mathbf{x}_0\in
\mathbb{C}^d$ from $m$ noisy intensity measurements $b_j=\left\lvert \langle
\mathbf{a}_j,\mathbf{x}_0 \rangle \right\rvert^2+\eta_j, \; j=1,\ldots,m$,
where $\mathbf{a}_j \in \mathbb{C}^d$ are known measurement vectors and
$\eta=(\eta_1,\ldots,\eta_m)^\top \in \mathbb{R}^m$ is a noise vector. A
commonly used model for estimating $\mathbf{x}_0$ is the intensity-based model
$\widehat{\mathbf{x}}:=\mbox{argmin}_{\mathbf{x} \in \mathbb{C}^d} \sum_{j=1}^m
\big(\left\lvert \langle \mathbf{a}_j,\mathbf{x} \rangle \right\rvert^2-b_j
\big)^2$. Although one has already developed many efficient algorithms to solve
the intensity-based model, there are very few results about its estimation
performance. In this paper, we focus on the estimation performance of the
intensity-based model and prove that the error bound satisfies $\min_{\theta\in
\mathbb{R}}\|\widehat{\mathbf{x}}-e^{i\theta}\mathbf{x}_0\|_2 \lesssim
\min\Big\{\frac{\sqrt{\|\eta\|_2}}{{m}^{1/4}}, \frac{\|\eta\|_2}{\|
\mathbf{x}_0\|_2 \cdot \sqrt{m}}\Big\}$ under the assumption of $m \gtrsim d$
and $\mathbf{a}_j, j=1,\ldots,m,$ being Gaussian random vectors. We also show
that the error bound is sharp. For the case where $\mathbf{x}_0$ is a
$s$-sparse signal, we present a similar result under the assumption of $m
\gtrsim s \log (ed/s)$. To the best of our knowledge, our results are the first
theoretical guarantees for the intensity-based model and its sparse version.
Our proofs employ Mendelson's small ball method which can deliver an effective
lower bound on a nonnegative empirical process.

14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg