Program

We revisit the so-called exit wave reconstruction problem in the variational setting. Here, exit wave reconstruction means to reconstruct the complex-valued electron wave in a transmission electron microscope (TEM) right before it passes the objective lens, i.e., the exit wave, from a focus series of real-valued TEM images.

This is a non-linear inverse problem that is a variant of the well-known phase retrieval problem. First, we discuss a classical variational approach to this setting. Here, existence of minimizers can be established in the functional space setting and numerical minimization is usually done with gradient descent-based approaches.

Applying the proximal gradient algorithm to a specific simplified version of this problem leads to an iterative algorithm well suited for deep unfolding. The latter is a technique that recasts iterative algorithms into neural networks and allows to introduce data-driven learning to a large class of classical iterative model-based approaches, creating new hybrid methods.

By extending a proof technique Behboodi, Rauhut and Schnoor proposed for a similar unfolded network but with a linear forward operator to our non-linear forward operator, we can show a generalization error bound for the resulting network. A key ingredient for the extension to our non-linear setting is the reinterpretation of the occurring non-linearities as proximal operators, which allows us to exploit firmly non-expansiveness.

In this course, I will give a brief overview of the physical models and mathematical techniques used to study the phase retrieval problem. The main objective will be to get everyone up to speed on the notation, abstractions, and results, and then to begin a discussion on what new mathematics could potentially be useful for helping tackle real-world phase retrieval problems.

Motivated by Grohs-Rathmair's Cheeger-type estimate for Gabor phase retrieval, we introduce a kernel Cheeger constant that quantifies connectedness relative to localization quantified in terms of a given reproducing kernel associated to a given (discrete or continuous) frame. This notion yields a clean stability certificate: we establish a unified lower bound over both real and complex fields, and in the real case also an upper bound, each in terms of the reciprocal kernel Cheeger constant.

Our framework treats finite- and infinite-dimensional settings uniformly and covers discrete, semi-discrete, and continuous measurement domains.

Based on joint work with Max Getter.

This talk presents a recent positive solution to the open problem of the existence of a uniformly discrete uniqueness set for the phase retrieval problem in Fock space. After introducing the problem and its motivation, I will place it in its broader mathematical context and outline the main ideas of the proof, which combines Remez inequalities with VC-dimension techniques.

The talk also discusses its formal verification in Lean. The results are based on joint work with Jaume de Dios Pont, Josef Greilhuber, and Mitchell Taylor.

Phase retrieval is a nonconvex inverse problem arising in a broad range of applications, including microscopy, diffractive imaging, and ptychography. In these applications, phaseless measurements of a signal are contaminated by various types of noise, including background noise and noise arising from the operating mode of detectors, which often measure variations in photon counts.

We study phase retrieval in the low-dose regime, where a low dose of radiation must be used to mitigate potential damage to the specimen, resulting in low observed particle counts. This regime is especially relevant to biological imaging and measurement systems with event-driven detectors, such as 4D-STEM, where detectors can record only the first particle arrival time.

In these settings, Poisson and Bernoulli noise models are more appropriate than the better-studied additive Gaussian noise model. In this talk, we will focus on Poisson and Bernoulli phase retrieval in the low-dose regime. We will discuss recovery guarantees for spectral initialization and quantify the trade-off between radiation dose per measurement and total sampling complexity.

For Poisson phase retrieval, we will further show that the maximum-likelihood estimator achieves a (nearly) minimax-optimal reconstruction rate and investigate how signal priors influence reconstruction in the low-dose regime by introducing a trade-off between total sampling complexity and the bias introduced by the signal model.

We present a Lean 4 library for the theory of Banach lattices, designed to support the systematic formalization of contemporary research in Banach lattices and related areas, such as Phase Retrieval.

To enable formalization at this scale, we employed Large Language Models under careful human supervision and planning. We also discuss the advantages and limitations of this approach compared with both fully automated and entirely manual formalization.

Finally, as initial use cases of the library, we present three research-level formalizations built on top of it.

The group orbit recovery problem seeks to estimate the orbit of a signal from multiple noisy observations, each transformed by a random group element. These models can be viewed as Gaussian mixture models whose centers are structured by group actions. Motivated by applications in modern structural biology, particularly single-particle cryo-electron microscopy (cryo-EM), we focus on the notoriously challenging high-noise regime.

The talk will cover two complementary aspects of orbit recovery:

1. Fundamental limits. We analyze the sample complexity and stability of orbit recovery, revealing deep connections between information-theoretic limits and the underlying algebraic structure of the group action.

2. Algorithms in the high-noise regime. We study the behavior of the expectation-maximization algorithm under high noise, explaining why it can become exceedingly slow, even in the vicinity of the true solution. We then present new algorithms that achieve the maximum-likelihood estimator significantly faster than EM, making large-scale orbit recovery problems substantially more practical.

A basic sequence (e_k) in a real Banach lattice X is called bibasic if, for any x in the closed linear span of {e_k : k ∈ N}, there exists a sequence (a_k) such that ∑_k=1^∞ a_ke_k converges to x both in norm and in order.

Equivalently, there exists a constant C so that, for any n and any real coefficients a₁, ..., a_n, the lattice supremum of the absolute values of the partial sums is bounded by C times the absolute value of the final sum.

Further, a basic sequence (e_k) is called absolute if a stronger condition holds -- namely, there exists a constant D that, for any n and any real coefficients, the lattice sum of |a_ke_k| is bounded by D times the absolute value of the corresponding linear combination.

Clearly any absolute sequence is bibasic, and the converse is false.

A Banach lattice is called an AM-space if it embeds into C(K) as a sublattice, for some Hausdorff compact K. It is known that a Banach lattice X is lattice isomorphic to an AM-space if and only if there exists a constant M so that, for any x₁, ..., x_n ∈ X, the lattice supremum of their absolute values is bounded by M times the corresponding lattice norm.

In such a Banach lattice, any unconditional basic sequence is clearly absolute. We show the opposite: if every absolute basic sequence in X is unconditional, then X is lattice isomorphic to an AM-space.

Further, any Banach lattice which is "far from being" either an AM-space or an L₁-space contains an unconditional basic sequence which is not bibasic.

Imaging methods based on phase retrieval have become quite popular in recent years, especially for X-ray and electron microscopy. However, the structure of the phase retrieval problems which are embedded in these methods can differ quite dramatically from the standard settings studied by mathematicians.

A great example of this is ptychography, the gold standard method for high-resolution X-ray and electron microscopy. The reconstruction problem for ptychography is built around the problem of phase retrieval from short-time Fourier transform magnitudes. Nonetheless, real ptychography data differs so much from this idealized setting as to be almost unrecognizable.

In this mini-course, I will explore these differences in detail using ptychography datasets from recent, well-known experiments as a guide. To put this data in context, we will derive the physical forward operators which capture these differences and discuss the algorithmic strategies used by experimentalists to account for them.

The final goal is to help identify possible generalized settings which may both be amenable to mathematical analysis while remaining relevant to the issues which experimentalists face in their work.

As has been suggested in recent works, Banach lattices provide a suitable abstract framework for modeling phase retrieval problems beyond the classical Hilbert space setting. In this talk, we review the characterization of phase retrieval and stable phase retrieval in real Banach lattices and extend these characterizations to the complex setting. We then apply this framework to C(K)-spaces, extending several results of Freeman, Oikhberg, Pineau and Taylor and solving a problem on the existence of quantitatively large infinite-dimensional subspaces that perform stable phase retrieval.

Machine learning algorithms are typically designed for Euclidean data, but many natural datasets come with symmetries: a group G of isometries acts on a Euclidean space V, and points in the same orbit represent the same object. That means the true data space is not V, but the orbit space V/G. Invariant machine learning represents this quotient by a G-invariant feature map into Euclidean space. For robustness, especially against adversarial examples, this feature map should be bilipschitz with respect to the quotient metric. Sadly, vanilla polynomial invariants fail to be bilipschitz, so we need to move beyond classical invariant theory. In this talk, we use ideas from phase retrieval to obtain low-distortion embeddings in a variety of settings, and we conclude with several open problems.

Ptychography is a diffraction imaging technique in which the sample is illuminated by a focused coherent light, called a window, producing a collection of far-field diffraction patterns. The sample is then recovered from the measurements computationally. Mathematically, it corresponds to a short-time Fourier phase retrieval problem. We focus on the Wigner distribution deconvolution method (WDD) for ptychographic recovery based on harmonic analytical tools. We show that it is a natural extension of the classical singular value decomposition for linear operators and discuss its implications for the uniqueness and stability of short-time Fourier phase retrieval.

Consider the one-dimensional Schrödinger equation with a real-valued potential iu_t + u_xx = Vu. In this talk, I will discuss a new approach for establishing phase retrieval for such equations in the natural setting where the solution lies in L². Using this method, we are able to answer a question posed by Jaming for a large class of potentials. In contrast to previous literature for this problem, the proof uses relatively simple techniques from the theory of nonlinear dispersive equations, such as local-smoothing, Strichartz estimates and unique continuation type arguments to overcome the limited regularity of the solution. The method is robust, and also applies to a large class of nonlinear Schrödinger equations, such as the cubic NLS equation. This is based on a joint work with João Ramos and Mitchell Taylor.

Affiliation: IMPA

The phase retrieval problem for the short-time Fourier transform is deeply routed in its applications beyond mathematics, such as in ptychography and in several other fields. Of particular mathematical interest, however, is the case when the window function is a Gaussian. In that case, the problem translates naturally into a problem of phase retrieval in the class of entire functions of the complex plane which are square integrable against e^{-π |z|2} - the so-called Bargmann-Fock space.

In this talk, we will survey several different results on such a problem, with a particular emphasis on recent progress on the problem of local phase retrieval in that space. We will discuss the recent results proving that, around certain functions - whose prime example is the constant function -, phase retrieval does hold, due to a geometric orthogonality condition yielded by circles. We will see how such a result connects to the problem of phase retrieval on domains, which when modified allows for a proof of the conjectured fact that local phase retrieval in the Fock space holds on any (regular enough) domain, provided we allow the constant to depend on the domain and one of the functions, and, time-permitting, we will discuss a sharp version of the aforementioned phase retrieval problem for squares.

Joint work with S. Bertolini, C. Bortolotto, J. de Dios Pont, B. Pineau and M. Taylor.

Cryo-EM is a Nobel Prize winning technology for determining 3-D biological molecular structures at high resolution. Reconstruction in cryo-EM is an inverse problem that involves many different fields of mathematics including statistical inference, optimization (convex and non-convex), numerical analysis, dimension reduction, representation theory, and information theory. In this mini-course, we will discuss the mathematical and statistical foundation underlying computational methods for 3-D reconstruction, focusing on the challenges of reconstructing small size molecules and the reconstruction of flexible molecules. We will also explore mathematical connections between cryo-EM reconstruction and phase retrieval.

Phase retrieval appears deceptively simple: recover a signal from intensity-only measurements. The loss of phase, however, makes the problem intrinsically nonconvex and has inspired a remarkably diverse collection of algorithms, from alternating projections and convex relaxations to spectral methods and modern nonconvex optimization.

In this mini-course, I will present phase retrieval algorithms, old and new, through a unified optimization lens. We will see how many seemingly different methods emerge naturally from different formulations of the same underlying problem, and how the geometry of these formulations shapes algorithm design. This viewpoint will connect classical methods with more recent developments and shed light on fundamental issues such as initialization, nonconvexity, priors, and the interplay between algorithms and measurement models.

The final part of the course will turn to the emerging role of AI in phase retrieval. We will explore how learning is changing the way phase retrieval and inverse problems are solved, from end-to-end neural reconstruction to deep generative priors and diffusion-based methods. We will examine different ways of combining learned models with optimization and physical measurement models, and their relative strengths and limitations. The broader theme is how classical optimization, physical modeling, and modern generative AI can come together to shape the next generation of phase retrieval algorithms and scientific image reconstruction more broadly.