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Seyedehsara Nayer

Publications and source records attributed to Seyedehsara Nayer.

5 recordsLinked to original sources
arXiv2021

Sample-Efficient Low Rank Phase Retrieval

This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an $n \times q$ rank-$r$ matrix $X^*$ from $y_k = |A_k^\top x^*_k|$, $k=1, 2,..., q$, when each $y_k$ is an m-length vector containing independent phaseless linear projections of $x^*_k$. The different matrices $A_k$ are i.i.d. and each contains i.i.d. standard Gaussian entries. We obtain an improved guarantee for AltMinLowRaP, which is an Alternating Minimization solution to LRPR that was introduced and studied in our recent work. As long as the right singular vectors of $X^*$ satisfy the incoherence assumption, we can show that the AltMinLowRaP estimate converges geometrically to $X^*$ if the total number of measurements $mq \gtrsim nr^2 (r + \log(1/ε))$. In addition, we also need $m \gtrsim max(r, \log q, \log n)$ because of the specific asymmetric nature of our problem. Compared to our recent work, we improve the sample complexity of the AltMin iterations by a factor of $r^2$, and that of the initialization by a factor of $r$. We also extend our result to the noisy case; we prove stability to corruption by small additive noise.

cs.IT↗
arXiv2020

Provable Low Rank Phase Retrieval

We study the Low Rank Phase Retrieval (LRPR) problem defined as follows: recover an $n \times q$ matrix $X^*$ of rank $r$ from a different and independent set of $m$ phaseless (magnitude-only) linear projections of each of its columns. To be precise, we need to recover $X^*$ from $y_k := |A_k{}' x^*_k|, k=1,2,\dots, q$ when the measurement matrices $A_k$ are mutually independent. Here $y_k$ is an $m$ length vector, $A_k$ is an $n \times m$ matrix, and $'$ denotes matrix transpose. The question is when can we solve LRPR with $m \ll n$? A reliable solution can enable fast and low-cost phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens. In this work, we develop the first provably correct approach for solving this LRPR problem. Our proposed algorithm, Alternating Minimization for Low-Rank Phase Retrieval (AltMinLowRaP), is an AltMin based solution and hence is also provably fast (converges geometrically). Our guarantee shows that AltMinLowRaP solves LRPR to $ε$ accuracy, with high probability, as long as $m q \ge C n r^4 \log(1/ε)$, the matrices $A_k$ contain i.i.d. standard Gaussian entries, and the right singular vectors of $X^*$ satisfy the incoherence assumption from matrix completion literature. Here $C$ is a numerical constant that only depends on the condition number of $X^*$ and on its incoherence parameter. Its time complexity is only $ C mq nr \log^2(1/ε)$. Since even the linear (with phase) version of the above problem is not fully solved, the above result is also the first complete solution and guarantee for the linear case. Finally, we also develop a simple extension of our results for the dynamic LRPR setting.

cs.LG↗
arXiv2018

Phaseless Subspace Tracking

This work takes the first steps towards solving the "phaseless subspace tracking" (PST) problem. PST involves recovering a time sequence of signals (or images) from phaseless linear projections of each signal under the following structural assumption: the signal sequence is generated from a much lower dimensional subspace (than the signal dimension) and this subspace can change over time, albeit gradually. It can be simply understood as a dynamic (time-varying subspace) extension of the low-rank phase retrieval problem studied in recent work.

cs.LG↗
arXiv2017

Low Rank Phase Retrieval

We develop two iterative algorithms for solving the low rank phase retrieval (LRPR) problem. LRPR refers to recovering a low-rank matrix $\X$ from magnitude-only (phaseless) measurements of random linear projections of its columns. Both methods consist of a spectral initialization step followed by an iterative algorithm to maximize the observed data likelihood. We obtain sample complexity bounds for our proposed initialization approach to provide a good approximation of the true $\X$. When the rank is low enough, these bounds are significantly lower than what existing single vector phase retrieval algorithms need. Via extensive experiments, we show that the same is also true for the proposed complete algorithms.

cs.IT↗