Algebraic Complexity and Neurovariety of Linear Convolutional Networks. (arXiv:2401.16613v1 [math.AG])

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 …

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