This paper studies the relations between the local minima of a cost function f and the stable equilibria of the gradient descent flow of f . In particular, it is shown that, under the assumption that… (More)

Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x 0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t)… (More)

Let f : R → R be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Lojasiewicz’s gradient inequality states there exist C > 0 and ρ ∈ (0, 1) such that |∇f | ≥ C|f | in a neighbourhood of… (More)

It is shown in this note that every invertible polynomial transformation of Rn of degree two has a rational inverse defined on the whole space R". The same is true for polynomial transformations of… (More)

We show that gradient trajectories of a definable (in an o-minimal structure) family of functions are of uniformly bounded length. We show that the length of a trajectory of the gradient of a… (More)

A function is called arc-analytic if it is real analytic on each real analytic arc. In real analytic geometry there are many examples of arc-analytic functions that are not real analytic. Arc… (More)