Neural Response Interpretation through the Lens of Critical Pathways

Is critical input information encoded in specific sparse pathways within the neural network? In this work, we discuss the problem of identifying these critical pathways and subsequently leverage them for interpreting the network's response to an input. The pruning objective -- selecting the smallest group of neurons for which the response remains equivalent to the original network -- has been previously proposed for identifying critical pathways. We demonstrate that sparse pathways derived from pruning do not necessarily encode critical input information. To ensure sparse pathways include critical fragments of the encoded input information, we propose pathway selection via neurons' contribution to the response. We proceed to explain how critical pathways can reveal critical input features. We prove that pathways selected via neuron contribution are locally linear (in an L2-ball), a property that we use for proposing a feature attribution method: "pathway gradient". We validate our interpretation method using mainstream evaluation experiments. The validation of pathway gradient interpretation method further confirms that selected pathways using neuron contributions correspond to critical input features. The code is publicly available.Comment: Accepted at CVPR 2021 (IEEE/CVF Conference on Computer Vision and Pattern Recognition

On pth roots in Henselian valued fields

Let v be a Henselian valuation of arbitrary rank of a field K of characteristic zero with value group G and residue field of characteristic p > 0. Suppose that K contains a primitive pth root of unity. It is well known that if d is an element of K with ν(d)>pν(p)/p-1 then (1 + d)1/P ε K.In this article we investigate whether is the smallest among all elements λ of G which have the property that whenever v(d)> λ,d ε K then 1 + d is a pth power in K

Difference polynomials and their generalizations

A well-known result of Ehrenfeucht states that a difference polynomial f(X)-g(Y) in two variables X, Y with complex coefficients is irreducible if the degrees of f and g are coprime. Panaitopol and Stefãnescu generalized this result, by giving an irreducibility condition for a larger class of polynomials called “generalized difference polynomials”. This paper gives an irreducibility criterion for more general polynomials, of which the criterion of Panaitopol and Stefãnescu is a special case

On limits of sequences of algebraic elements over a complete field

Let K be a complete field with respect to a real non-trivial valuation v, and ν̅be the extension of v to an algebraic closure K̅ of K. A well-known result of Ostrowski asserts that the limit of a Cauchy sequence of elements of K̅ does not always belong to K̅ unless K̅is a finite extension of K. In this paper, it is shown that when a Cauchy sequence { bn} of elements of K̅ is such that the sequence { [K(bn): K] } of degrees of the extensions K(bn)/K does not tend to infinity as n approaches infinity, then {bn}has a limit in K̅.We also give a characterization of those Cauchy sequences {bn} of elements of K̅whose limit is not in K̅,which generalizes a result of Alexandru, Popescu and Zaharescu

A Characterization of Krasner's Constant

Let v be a henselian valuation of any rank of a eld K with value group G and v be its unique prolongation to a xed algebraic closure K of K: For an element of K\K; which is separable over K; let !K () denote the well known Krasner's constant given by maxfv( 6= runs over K conjugates of g: In 1946, Krasner proved that if belonging to K is such that v( ) > !K (); then K() K( ): In this paper, we investigate whether !K () is the smallest among all the elements of the divisible closure of G which have the property that whenever v( ) > ; 2 K; then K() K( )

Difference Polynomials and Their Generalizations

this paper, we study irreducibility conditions of more general polynomials given by F (X; Y ) = cX ; c 2 k ; e 1; such that there exists t; 1 t e satisfying deg Y F (X; Y ) = degP t (Y ) = d and degP i (Y ) < di=t for i 6= t; 1 i e: Such a polynomial F (X; Y ) will be referred to as a quasi-dierence polynomial of the type (d; t) with respect to X: In 1990, L. Panaitopol and D. Stefanescu proved that a quasi-dierence polynomial of the type (d; t) with d and t coprime is irreducible over k[X] (cf. [5, Theorem 6]). In this direction, we go further and give an irreducibility criterion for a quasi-dierence polynomial of the type (d; t) with d and t not necessarily coprime, of which the criterion of Panaitopol and Stefanescu is a special case. Our method of proof is dierent from the one employed in [2] and [5], and is based on the idea of the proof of the Generalized Eisenstein's Irreducibility Criterion given in [4