arXiv subjects

Josh Katz

Publications and source records attributed to Josh Katz.

7 recordsLinked to original sources

arXiv2026

Projected multi-reference alignment

Motivated by structural biology applications, we study the projected multi-reference alignment (MRA) model, in which an unknown signal is observed through noisy samples, each generated by applying a random cyclic shift followed by a fixed projection. The projection merges reflection-symmetric index pairs, thereby discarding orientation information. The goal is to recover the dihedral orbit of the signal. We prove that in the high-noise regime, the first three moments of the projected observations determine a generic dihedral orbit. The main mechanism is a reduction, at the moment level, from projected MRA to the reflection-invariant phase-coupling structure of dihedral MRA. In Fourier-cosine coordinates adapted to the projection, the first moment determines the mean component, the second moment determines the Fourier magnitudes, and selected third moments yield the cosine phase-coupling relations appearing in the dihedral bispectrum. These relations lead to a constructive recovery scheme from moments up to order three. We complement the population theory with finite-sample experiments comparing expectation--maximization (EM), direct moment optimization, and direct Fourier-cosine moment optimization. The results show that, in the high-noise regime, both EM and direct moment optimization are consistent with the predicted third-moment sample-complexity scaling $n \gtrsim \sigma^6$, where $n$ is the number of observations and $\sigma^2$ is the noise variance.

eess.SP↗

arXiv2026

Provable orbit recovery over SO(3) from the non-uniform second moment

We study the recovery of an unknown three-dimensional band-limited signal from multiple noisy observations that are randomly rotated by latent elements of SO(3), where the rotations are drawn from an unknown, non-uniform distribution. Because the rotations are unobserved, only the signal orbit under the rotation group can be recovered. We show that the signal orbit and the rotation distribution are jointly identifiable from the first and second moments. This yields an improved high-noise sample complexity that scales quadratically with the noise variance, rather than cubically as in the uniform-rotation case. We further develop a provable, computationally efficient reconstruction algorithm that recovers the 3-D signal by successively solving a sequence of well-conditioned linear systems. The algorithm is validated through extensive numerical experiments. Our results provide a principled and tractable framework for high-noise 3-D orbit recovery, with potential relevance to cryo-electron microscopy and cryo-electron tomography modeling, where molecules are observed in unknown orientations.

eess.SP↗

arXiv2026

Unitary Invariants of the Finite Heisenberg Group

Polynomial invariants of a group action often appear only in high degree, and in many representations the invariant ring imposes severe degree constraints before any nontrivial invariants can occur. In contrast, the larger class of unitary invariants -- polynomials in both the variables and their conjugates -- typically exhibits very different behavior, and their separating power is comparatively unexplored. We highlight this contrast in the setting of the finite Heisenberg group $H_N$. Although the polynomial invariant ring $\mathbb{C}[V]^{H_N}$ contains no nontrivial elements below degree $N$, we show that degree-six unitary invariants are already sufficient to separate generic $H_N$-orbits up to a global phase factor. These invariants arise from cubic equations involving the magnitudes of a vector and its discrete Fourier transform. A single polynomial invariant in degree $N$ then resolves the remaining global phase, yielding full generic orbit separation. Our proof utilizes fundamental results from phase retrieval. Along the way we will also explore the utility of unitary invariants in obtaining improved degree bounds for representations of cyclic groups. This paper provides a concrete example in which the minimal separating degree for unitary invariants is dramatically lower than the minimal degree for polynomial invariants.

math.RT↗

arXiv2025

Orbit recovery for spherical functions

Orbit recovery is a central problem in both mathematics and applied sciences, with important applications to structural biology. This paper focuses on recovering generic orbits of functions on ${\mathbb R}^{n}$ and the sphere $S^{n-1}$ under the rotation action of $SO(n)$. Specifically, we demonstrate that invariants of degree three (called the bispectrum) suffice to recover generic orbits of functions in finite-dimensional approximations of $L^2({\mathbb R}^n)$ obtained by band-limiting the spherical component and discretizing the radial direction. In particular, our main result explicitly bounds the number of samples in the radial direction required for recovery from the degree three invariants. From an application perspective, the most important case is $SO(3)$, which arises in many scientific fields, and in particular, plays a central role in leading structural biology applications such as cryo-electron tomography and cryo-electron microscopy. Our result for $SO(3)$ states that considering three spherical shells (i.e., samples in the radial direction) is sufficient to recover generic orbits, which verifies an implicit conjecture made in a paper of Bandeira et al. Our proof technique provides an explicit, computationally efficient algorithm to recover the signal by successively solving systems of linear equations. We implemented this algorithm and demonstrated its effectiveness on three protein structures.

math.NA↗

arXiv2025

Polarization algebras and the geometry of commuting varieties

We prove a reduced version of the Chevalley restriction conjecture on the commuting scheme posed by T.H. Chen and B.C. Ng\^o, extending the results of Hunziker for classical groups. In particular, we prove that for any connected reductive group, the ring of $G$-invariant functions on the commuting variety restricts to an isomorphism with the invariants of the d-fold product of a Cartan subalgebra under the Weyl group $k[\mathfrak{t}^d]^W$. The full conjecture implies that this isomorphism extends to the ring of $G$-invariants on the non-reduced commuting scheme, $k[\mathfrak{C}_{\mathfrak{g}}^d]^G$ (hence the invariant ring is nilpotent free). We then prove an analogous restriction theorem for general polar representations which we apply to resolve an important case of a conjecture posed by Bulois, C Lehn, M Lehn and Terpereau about symplectic reductions of $\theta$-representations. Throughout this work, we focus on the connection between the invariant subring generated by polarizations and commutativity.

math.RT↗

arXiv2025

Orbit recovery from invariants of low degree in representations of finite groups

Motivated by applications to equivariant neural networks and cryo-electron microscopy we consider the problem of recovering the generic orbit in a representation of a finite group from invariants of low degree. The main result proved here is that invariants of degree at most three separate generic orbits in the regular representation of a finite group defined over any infinite field. This answers a question posed in a 2023 ACHA paper of Bandeira et. al. We also discuss this problem for subregular representations of the dihedral and symmetric groups.

math.RT↗

arXiv2024

The reflection invariant bispectrum: signal recovery in the dihedral model

We study the problem of signal recovery in the dihedral multi-reference alignment (MRA) model, where a signal is observed under random actions of the dihedral group and corrupted by additive noise. While previous has shown that cyclic invariants of degree three (the bispectrum) suffice to recover generic signals up to circular shift, the dihedral setting introduces new challenges due to the groups non-abelian structure. In particular reflections prevent the diagonalization of the third moment tensor in the Fourier basis, making classical bispectrum techniques inapplicable. In this work we prove that the orbit of the generic signal in the $n$-dimensional standard representation of the then $2n$-element dihedral group $D_{n}$ is uniquely determined by invariant tensors of degree at most three. This resolves an open question in the literature and establishes that the sample complexity for dihedral MRA with uniform distribution is $\omega(\sigma^6)$ matching the cyclic case. While frequency marching becomes computationally impractical in the dihedral setting, we show numerically that a simple optimization algorithm reliably recovers the signal from third order moments, even with random initialization.

math.AC↗