We describe a method to recover the underlying parametrization of scattered data (mi) lying on a manifold M embedded in high-dimensional Euclidean space. The method, Hessian-based Locally Linear Embedding (HLLE), derives from a conceptual framework of Local Isometry in which the manifold M , viewed as a Riemannian submanifold of the ambient Euclidean space Rn, is locally isometric to an open, c...

This paper is a short version of some joint work with Stefan Haller. It describes the structure of "smooth manifold with corners" on the space of possibly broken instantons of a generic smooth vector field. The result is stated in Theorem 1.4. CONTENTS 1. The results 1 2. Some basic ODE 7 3. Elementary differential topology of smooth manifolds with corners 9 4. Proof of the main theorem 10 Refe...

Let Θ(M,K) denote the 2-loop piece of (the logarithm of) the LMO invariant of a knot K in M , a ZHS . Forgetting the knot (by which we mean setting diagrams with legs to zero) specialises Θ(M,K) to λ(M), Casson’s invariant. This note describes an extension of Casson’s surgery formula for his invariant to Θ(M,K). To be precise, we describe the effect on Θ(M,K) of a surgery on a knot which togeth...

Let M = (E,F) be a matroid on a set E, and B one of its bases. A closed set θ ⊆ E is saturated with respect to B when |θ ∩B| = r(θ), where r(θ) is the rank of θ. The collection of subsets I of E such that |I ∩ θ| ≤ r(θ) for every closed saturated set θ turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by MB . In this paper we prove that a grap...

We show that as T→∞, for all t∈[T,2T] outside of a set measure o(T), ∫−logθTlogθT|ζ(1 2+it+ih)|βdh=(logT)fθ(β)+o(1), some explicit exponent fθ(β), where θ>−1 and β>0. This proves an extended version conjecture Fyodorov Keating (Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20120503, 32). In particular, it shows that, θ>−1, the moments exhibit phase transition at critical ...

P~ θ : V 7→ [0, 1], where ~ θ is an element of the m-dimensional probability simplex. Hence the probability assigned to a single term vj is defined as: P~ θ (vj) def = θ[j]. Also recall from the previous lecture that the Kullback–Leibler (KL) divergence between two probability distributions P~ θ and P~ θ′ , i.e. the expected log-likelihood ratio with respect to P~ θ, is defined as: D(P~ θ ‖P~ θ...

In [1], it was conjectured that the permanent of a P-lifting θ of a matrix θ of degree M is less than or equal to the M th power of the permanent perm(θ), i.e., perm(θ) 6 perm(θ) and, consequently, that the degree-M Bethe permanent permM,B(θ) of a matrix θ is less than or equal to the permanent perm(θ) of θ, i.e., permM,B(θ) 6 perm(θ). In this paper, we prove these related conjectures and show ...

Strong convexity can be characterized in a number of ways. The following facts provide some conditions that are equivalent to Definition 1. For the 2-norm, Fact 2 means that the minimum eigenvalue of the Hessian is lower-bounded by κ. Recall that˜µ(θ) and˜µ(θ) are the gradients of˜Φ(θ) and˜Φ(θ), respectively. Since the conjugate function, ˜ Φ * , is assumed to be κ-strongly convex, we have via ...

Many practitioners who use EM and related algorithms complain that they are sometimes slow. When does this happen, and what can be done about it? In this paper, we study the general class of bound optimization algorithms – including EM, Iterative Scaling, Non-negative Matrix Factorization, CCCP – and their relationship to direct optimization algorithms such as gradientbased methods for paramete...