In this paper, we study linear convolutional networks with one-dimensional filters and arbitrary strides. The neuromanifold of such a network is a semialgebraic set, represented by a space of polynomials admitting specific factorizations. Introducing a recursive algorithm, we generate polynomial equations whose common zero locus corresponds to the Zariski closure of the corresponding neuromanifold. Furthermore, we explore the algebraic complexity of training these networks employing tools from metric algebraic geometry. Our findings reveal that the number of all complex critical points in the optimization of such a network is equal to the generic Euclidean distance degree of a Segre variety. Notably, this count significantly surpasses the number of critical points encountered in the training of a fully connected linear network with the same number of parameters.

We study convolutional neural networks with monomial activation functions. Specifically, we prove that their parameterization map is regular and is an isomorphism almost everywhere, up to rescaling the filters. By leveraging on tools from algebraic geometry, we explore the geometric properties of the image in function space of this map - typically referred to as neuromanifold. In particular, we compute the dimension and the degree of the neuromanifold, which measure the expressivity of the model, and describe its singularities. Moreover, for a generic large dataset, we derive an explicit formula that quantifies the number of critical points arising in the optimization of a regression loss.

We study the geometry of linear networks with one-dimensional convolutional layers. The function spaces of these networks can be identified with semialgebraic families of polynomials admitting sparse factorizations. We analyze the impact of the network’s architecture on the function space’s dimension, boundary, and singular points. We also describe the critical points of the network’s parameterization map. Furthermore, we study the optimization problem of training a network with the squared error loss. We prove that for architectures where all strides are larger than one and generic data, the nonzero critical points of that optimization problem are smooth interior points of the function space. This property is known to be false for dense linear networks and linear convolutional networks with stride one.

Let G be a Frobenius group. Using character theory, it is proved that the Frobenius kernel of G is a normal subgroup of G, which is well-known as a Frobenius theorem. There is no known character-free proof for Frobenius theorem. In this note, we prove it, by assuming that Frobenius groups are non-simple. Also, we prove that whether K is a subgroup of G or not, Sylow 2-subgroups of G are either cyclic or generalized quaternion group. Also by assuming some additional arithmetical hypothesis on G we prove Frobenius theorem. We should mention that our proof is character-free.

Multireference alignment (MRA) refers to the problem of recovering a signal from noisy samples subject to random circular shifts. Expectation--maximization (EM) and variational approaches use statistical modeling to achieve high accuracy at the cost of solving computationally expensive optimization problems. The method of moments, instead, achieves fast reconstructions by utilizing the power spectrum and bispectrum to determine the signal up to shift. Our approach combines the two philosophies by viewing the power spectrum as a manifold on which to constrain the signal. We then maximize the data likelihood function on this manifold with a gradient-based approach to estimate the true signal. Algorithmically, our method involves iterating between template alignment and projections onto the manifold. The method offers increased speed compared to EM and demonstrates improved accuracy over bispectrum-based methods.