A rhapsody in multiple dimensions

Good morning. My name is Alex Bronstein, and I am Freddy Bruckstein’s grandson. Academically speaking, of course — but a proud one. I’m honored to be here, and I admit that the format I’ve chosen today is a little unorthodox, as the title may suggest.

Before we begin, let me tell you how I first met Freddy — some twenty-five years ago, when I was a fresh graduate student at the Technion. One day Freddy summoned me and my brother Michael to his office. An important full professor demanding to see you — terrifying. I spent a sleepless night rehearsing every tricky question about geometry and calculus he might ask. Instead, when we arrived, he opened a website with verses by the Romanian poet Mihai Eminescu, and recited this poem aloud.

Freddy — would you, like back then, read it for us?

The poem is about a distant star whose light still reaches us across the centuries even though the star itself may have long since burned out — paralleling it to the light of an extinguished love that still continues to illuminate our life. “This is a poem about Einstein’s relativity,” Freddy declared. “I hear you and Michael have been translating poems into Italian — would you do this one too?”

That was quintessential Freddy: physics and lyrics in the same breath. That is how I met him, half a lifetime ago. It took me some years — and some life experience — to understand how true this poem really is. But I haven’t come here to talk about distant light sources or extinguished loves; those, perhaps, belong in a novel I may yet write someday. What I want to do today is recount how the afterglow of one idea I encountered in Freddy’s school has illuminated my work to this very day.

Thinking about what to show today made me reflect on a journey that has now lasted more than half my lifetime. A little like another poet who, nel mezzo del cammin della sua vita, found himself within a forest dark. Today we’d call it a mid-life crisis. Dante took his readers through a sequence of landscapes and, at the end, was able to rebehold the stars. I’d like to do something similar today — please don’t take the comparison too seriously. Time is short, so I won’t show you details. But that, too, I learned from Freddy: he came to class with handwritten, hand-drawn slides, and instead of mathematical rigor mortis, he used the math as illustration. I’ll try to do the same.

Let me open with the canon — the canonical form, that is. When I started my PhD with Ronny Kimmel, Freddy’s academic son, I was struck by his work with Asi Elad on the similarity of deformable shapes. Imagine an articulated object like my hand: different articulations don’t look the same, but the intrinsic geometry doesn’t change. A little ant living on the surface notices nothing when I bend my fingers — and yes, Freddy, I managed to slip an ant in.

Elad and Kimmel proposed straightening the geodesic paths and re-embedding them in Euclidean space, so that pairwise distances become straight lines, the way a fly would measure them. The messy problem of comparing deformable shapes — with so many degrees of freedom — collapsed into the much simpler problem of aligning two rigid 3D objects, with just six. Computationally, that boils down to multidimensional scaling, MDS for short: find a configuration of points whose Euclidean distances match a given distance matrix. This will be our theme, and everything that follows will sound like a variation.

Michael and I — yes, my evil identical twin and lifelong collaborator — used canonical forms for expression-invariant face recognition. The press, with its usual restraint, declared it was game over for the bad guys. We even built a full system: a fast C++ implementation of least-squares MDS, and our own coded-light 3D camera. A few years later that camera became Intel’s RealSense — only smaller and cheaper.

But Elad and Kimmel’s canonical forms came with a structural limit: MDS into Euclidean space always introduces some distortion, and that distortion blurred our ability to tell nearly-isometric shapes apart. We had to think harder.

The trick was to turn the foe into a friend. Instead of embedding both shapes into a fixed flat space, embed one shape directly into the other. Then the embedding distortion is no longer a nuisance — it measures the dissimilarity of the two metrics. And as a bonus, you get the minimum-distortion correspondence for free.

This turned out to be related to a classical extension of the Hausdorff distance by Mikhail Gromov. The idea: take two metric spaces, glue them, complete the metric across the glue, and minimize the Hausdorff distance over all valid extensions. Mathematically, that is identical to embedding one shape into the other with minimum distortion. Generalized MDS turned a metric-geometry existence theorem into a working algorithm. The bottleneck, of course, was that we now had to optimize over point-wise correspondences — and these maps between manifolds can be quite ugly.

The way out came from Maks Ovsjanikov, who did his PhD with Leo Guibas at Stanford. His proposal was to lift the problem: don’t look for a map between points on the two surfaces; look for an operator mapping functions on one surface to functions on the other. By a trivial property of function composition, this functional map is linear — and once it’s linear, you can do linear algebra on it.

Like any linear operator, a functional map can be represented in a basis — and the natural choice is the eigenfunctions of the Laplace–Beltrami operator, the manifold version of the Fourier basis. Together with my academic son Or Litany — Freddy’s great-grandchild, now a Technion professor sitting right here, I’m proud of you, Or — and Ema Rodolà, now a professor at La Sapienza in Rome (proud of him as well), we showed that one can do better still: jointly diagonalize the two Laplacians and you obtain a common spectral language across the shapes, even when one or both are only partially observed.

The catch: this magic requires you to start from some pairs of corresponding functions. If you have them, the problem is easy. But what if you don’t? It turns out the joint diagonalization can be solved by matching heat kernels — the Green functions of the diffusion equation on each manifold. The kernels carry exactly the same Gram structure as the pairwise geodesic distances. So this is, once again, an MDS problem in disguise. That line of work brought Or and Ema well-deserved recognition, and for me it closed the chapter — the deformable-shape correspondence problem became, practically, solved.

Let me show you one application of these ideas that I take very close to the heart — quite literally. About a decade ago, I had the privilege of consulting for a company called EP Dynamics, whose R&D was led by none other than Asi Elad — yes, the same Elad of the original canonical forms. It seems the paths of life have a fixed point. They were building a system for radio-frequency ablation of cardiac arrhythmias: a catheter threaded through the femoral artery up into the heart, and to perform the ablation safely you need to know, with millimeter precision, where the catheter is and what tissue it is touching.

Their setup placed conductive pads on the body, established a potential gradient, and read the local voltage from electrodes on the catheter — three axes, multiplexed at different frequencies. The trouble: the dielectric properties of biological tissue are wildly anisotropic and inhomogeneous, so the catheter sees a deeply distorted world. But there was one shard of rigidity. The catheter tip carried four electrodes whose physical inter-electrode distances were known and constant.

Online, we collected the electrode trajectories in voltage space, incrementally built a set of manifold harmonics, and represented in that basis the deformation that converts voltages back into anatomy — the deformation that preserves the four rigid distances at every instant. A spectral flavor of MDS, online, supervised by four numbers. The accuracy reached CT-quality mapping. It was an enlightening experience — especially the pre-clinical trials on pigs, after which I stopped eating meat. The company was acquired by Philips for half a billion dollars.

Notice the inversion that has just taken place. Until now, the manifold was given — the shape, the mesh, the surface; we already had it. With the catheter, we learned the manifold itself, from a handful of rigid distances. The same inversion now happens one level up.

Armed with this spectral toolkit and excited about kernel matching, with two more academic children — Sanketh and Amit — we turned to a more abstract setting: matrix completion. The classical Netflix problem. A huge users-by-films matrix, sparsely observed because each viewer rates only a handful of titles. Interpolate the rest.

Assume rows and columns each carry a graph structure — Michael probably has tastes similar to mine, so we’re close on the row-graph. Then a matrix is just a function on the product of two graphs, and the natural regularizer is a Dirichlet energy on that product. But the graphs themselves are unknown. Just as the catheter learned the chamber, here we learn the graphs and complete the matrix simultaneously — by solving a joint diagonalization problem, the very same one we used for shapes. The work also has lovely connections to deep matrix factorization and implicit rank regularization by stochastic gradient — but, as Fermat would say, those don’t fit in the margin of this page.

The problems so far have all been static — there is no time in shape correspondence or matrix completion. When my academic son Yonathan and I turned to dynamical systems, I was surprised to discover that what we and others call the functional map is, almost word for word, the Koopman operator — proposed by Bernard O. Koopman a century ago. Both are linear operators on functions. The Koopman operator represents the time propagator of a dynamical system as a linear operator on observable functions: it tells you how each observable evolves under one step of the dynamics. The same kind of object as a functional map — only that the “correspondence” is now between a system and itself one tick later.

Koopman operators on real data suffer from a partial-observability problem: a single trajectory only samples each observable at one point. So we asked: what if we observe many short trajectories from related dynamical systems? Cardiac data is exactly such a case — catching an arrhythmia from a normal-looking ECG can benefit from sharing dynamical modes across patients. We jointly approximated the Koopman operators of multiple patients with a common set of dynamical modes — coupled quasi-harmonic bases for time. From a short signal, the system could predict that a patient would later have an arrhythmia, even when their current rhythm looked perfectly normal to a cardiologist.

Some six years ago, my trajectory pivoted again — after meeting Ailie Marx, a structural biologist who introduced me to the wondrous world of proteins, the nano-machines that make up and run the edifice of Life. When I published my first protein paper, one of my PhD students remarked that I had finally moved from RealSense to real science. Poor fellow — it took him only ten years to finish his thesis.

You’ve surely heard of AlphaFold, which predicts how an amino-acid sequence folds in three dimensions — Nobel prize in 2024, transformative for the field. But here is a secret structural biologists have known for seventy years: the static structure in the picture is a lie. Even when hemoglobin’s structure was first resolved, it was clear the picture didn’t tell the full story — there was no path for the heme to enter the protein. Dynamics is essential. If a protein doesn’t move, it’s dead.

NMR — nuclear magnetic resonance, the same physics as the MRI machines you find in any clinic, only with the word nuclear dropped because it scares patients — can probe these dynamics by measuring inter-atomic distances. But what NMR sees is an ensemble average over millions of protein copies in different conformations. There is typically no single Euclidean structure that realizes the observed averages. So we are back to our favorite dish: an MDS problem, this time in probabilistic flavor. We sample from the posterior over conformational ensembles by steering AlphaFold’s diffusion sampler with the gradient of the observation likelihood — which is, at heart, a probabilistic MDS loss. A few minutes of compute produces ensembles that fit the experiment far better than any single structure ever could.

Proteins, together with other biomolecules, make up and run the edifice of living cells. Understanding what happens at the single-cell level — using molecular probes — lets us explain why cells can be as diverse as muscles and neurons while sharing exactly the same genome, and how that diversity develops. Each probe reveals only part of the cell’s state, so we typically need several: chromatin accessibility, transcriptomics, proteomics. The measurement is destructive — you cannot read all three from the same cell. So we end up with separate point clouds in different abstract high-dimensional spaces, sampled from the same underlying biology but never from the same individual cell. We’re back to a correspondence problem — without correspondences.

Different modalities are merely different views of the same underlying biological object, so they should share metric structures. That makes this a Gromov–Wasserstein problem — but quadratic assignment scales poorly. Our trick: embed both modalities into a common metric space — canonical forms again, only with more dimensions — using a pair of neural networks chosen so that the linear assignment of the embedded points minimizes the quadratic assignment loss. A bilevel MDS problem, with the bonus that the embeddings are inductive and extend to new cells for free. It scales to many millions of points, and reads as an amortized solution of GW by sequential linearization.

One last twist: a distance is not the only kind of pairwise structure that matters. We can replace raw distances with their ranks — made differentiable through optimal transport — which gives a wider family of GW-like problems that are invariant to any monotone transformation of the metric. Even if our distance functions are wrong up to a monotone reparameterization, the alignment still recovers. Robustness, courtesy of OT.

As you can see — exactly as in Eminescu’s poem — the afterglow of that one MDS idea has continued to permeate my work twenty-five years later. I cannot conclude without returning to the star.

I did translate the poem into Italian back then, of course. But in the years since I have changed too many addresses, and that text was lost in translation. So let me offer a present-day attempt. I have, I hope, a slightly better ear for Italian, a better notion of Einstein’s relativity, and a wiser perspective on remote light sources and extinguished loves. Let me read you a low-distortion translation-invariant map of La steaua into Italian.Eminescu’s La steaua and the translation, La stella, are on the translations page.

Thank you, Freddy — for everything. Proud to be your grandchild, proud to belong to your school. And thank you, ladies and gentlemen, for your attention.