Can you recover a function $f$ from its absolute value $|f|$? Recovering $f$ exactly is impossible: if $\lambda \neq 1$ is a complex scalar with modulus $1$, then $|f| = |\lambda f|$ although $f \neq \lambda f$. Thus, one rather asks whether $f$ can be recovered up to multiplication by a unimodular scalar. If this is possible for arbitrary $f$ in a subspace $E \subset L^2(\mathbb R)$, we say $E$ does phase retrieval.

Further, we can ask whether the recovery can be done in a controlled way – stably. Given a subspace $E \subset L^2(\mathbb{R})$ and $f, g \in E$, is there a constant $C > 0$ such that

If so, we say $E$ does stable phase retrieval.

A cute question… #

The question of stable phase retrieval is easy to state but hard to answer, much like combinatorics questions. In fact, I wrote a 40-page semester paper about aspects of this question; stable phase retrieval a highly active research field within harmonic analysis. This blog post is intended as a mathematical apéritif, so you know which bits to read.

The semester paper is split in two parts: Part I surveys prior work; Part II is a modest attempt to extend prior work. We drop assumptions, simplify arguments and modify proofs for other problems.

My co-supervisor, Dr. Mitchell Taylor, who authored the 2022 paper on the first infinite-dimensional subspaces of complex-valued $L^2(\mathbb{R})$ doing stable phase retrieval, shared ideas on the extensions presented in Part I. These are the ideas developed in Part II.

I would like to thank both my supervisors, Dr. Mitchell Taylor and Prof. Dr. Alessio Figalli for making this project possible. I also want to express my sincere gratitude to Dr. Taylor for valuable discussions – both on the mathematical content and on the research process itself.

Part I #

Part I kicks off with some preliminaries from Fourier series and Banach lattices. Lacunary Fourier series are Fourier series with sparsity conditions on the frequencies appearing in the Fourier expansion. Apparently you can say a lot about their $L^p([0, 1])$ norms – for instance, they’re all equivalent. The statement of Nazarov’s inequality, which concerns another kind of ‘sparse’ Fourier series, was nowhere to be found except in his doctoral thesis written in Russian, and is another highlight from this subsection. We also develop some Banach lattice theory. Banach lattices are Banach spaces with a lattice structure which makes them the ‘right’ setting for studying stable phase retrieval.

Then we outline the problem of stable phase retrieval, supplying many examples and non-examples of subspaces doing stable phase retrieval. My favourite example of a subspace failing stable phase retrieval is the linear span of characteristic functions on $[0, 1]$ and $[2, 3]$; in my notation, that’s $\langle 1_{[0, 1]}, 1_{[2, 3]}\rangle$. To see why this subspace fails stable phase retrieval, notice that $u := 1_{[0, 1]} + 1_{[2, 3]}$ and $v := 1_{[0, 1]} - 1_{[2, 3]}$ have the same modulus, though there’s no $\lambda$ on the unit circle such that $u = \lambda v$.

We then proceed to discuss some striking results about stable phase retrieval in Banach lattices, justifying the earlier slogan about Banach lattices being good for phase retrieval. Just to wet your appetite: subspaces of Banach lattices doing stable phase retrieval admit an easy characterisation, and for finite-dimensional subspaces of Banach lattices, one can show that phase retrieval is automatically stable.

The final section of part one focuses on the first constructions of infinite-dimensional subspaces of $L^2(\mathbb{R})$ doing stable phase retrieval from 2022. We present two results, the first due to R. Calderbank, I. Daubechies, D. Freeman and N. Freeman (subspaces of real-valued $L^2(\mathbb{R})$); the second due to M. Christ, B. Pineau, T. Oikberg and M. Taylor (subspaces of complex-valued $L^2(\mathbb{R})$). The first construction involves a construction with orthonormal iid random variables and characteristic functions, while the second construction involves bases satisfying four well-chosen conditions.

Part II #

Part two is a modest attempt to extend the work on infinite-dimensional subspaces of $L^2(\mathbb R)$ doing stable phase retrieval.

First, we describe one way of weakening the conditions in the Christ-Pineau-Taylor result. One of the four conditions says that a set of functions should be orthogonal. However, because stable phase retrieval is an approximate condition (we want a ‘$\le C$’, not a ‘$=$’), and orthogonality is a condition of exact equality, we try relaxing the orthogonality to something like ‘almost orthogonality’. This leads one to consider so-called Riesz bases, which, intuitively, are ‘almost orthogonal’ sequences. We argue that the same proof goes through for Riesz bases too.

Second, we simplify an argument in the Christ-Pineau-Taylor paper, using a reduction suggested by a subsequent paper. Namely, one can show that orthogonal vectors constitute a kind of worst-case scenario for stable phase retrieval: if functions $f$ and $g$ break the stable phase retrieval inequality, then you can extract a pair of orthogonal vectors $f'$ and $g'$ violating the same inequality. So, by contraposition, it’s enough showing arbitrary orthogonal vectors from your subspace $E$ satisfy the stable phase retrieval inequality to conclude that the $E$ does stable phase retrieval as a whole.

Third, we adapt the proof of Calderbank-Daubechies-Freeman-Freeman to the setting of Pauli-stable phase retrieval. A subspace $E \subset L^2([0, 1])$ does Pauli-stable phase retrieval if, for $f, g \in E \subset L^2([0, 1])$, we have

where the Fourier transform of an $L^2([0, 1])$ function is taken to be its Fourier series (which is in $\ell^2$, says Parseval).

In the report, we build subspaces from sine functions where the Fourier transforms play the same role as the characteristic functions in the original paper. Adapting their proof, then, leads to a result about infinite-dimensional subspaces of real-valued $L^2([0, 1])$ doing Pauli-stable phase retrieval.

The main difficulty in this proof was finding a workaround for independence – the original construction involves iid random variables, but our building blocks are sine functions. To have the functions in our subspace satisfy the same kinds of bounds, one can place sparsity conditions on the Fourier frequencies. And yes – this is where Nazarov’s inequality comes in.

This should be enough for an apéritif. I invite you to have a closer look at the report – the actual buffet. I had plenty of fun working on this project, and I hope this comes through in the full report.