We propose novel approaches to cast the widely-used family of Dynamic Time Warping (DTW) distances and similarities as positive definite kernels for time series. To this effect, we provide new theoretical insights on the family of Global Alignment kernels introduced by Cuturi et al. (2007) and propose alternative kernels which are both positive definite and faster to compute. We provide experimental evidence that these alternatives are both faster and more efficient in classification tasks than other kernels based on the DTW formalism.

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Fast Global Alignment Kernels

Dyck path enumeration

Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

The history of q-calculus and a new method

Let S be a subset of {-1,0,1}^2 not containing (0,0) We address the enumeration of plane lattice walks with steps in S, that start from (0,0) and always remain in the first quadrant A priori, there are 2^8 problems of this type, but some are trivial Some others are equivalent to a model of walks confined to a half-plane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions We focus on the remaining cases, and show that there are 79 inherently different problems to study To each of them, we associate a group G of birational transformations We show that this group is finite in exactly 23 cases We present a unified way of solving 22 of the 23 models associated with a finite group For each of them, the generating function is found to be D-finite The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al to have an algebraic (and hence D-finite) solution We conjecture that the remaining 56 models, associated with an infinite group, have a non-D-finite generating function Our approach allows us to recover and refine some known results, and also to obtain new results For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function

Walks with small steps in the quarter plane

Noncrossing partitions

The central Delannoy numbers, (dn)n‚0 = 1;3;13;63;321;1683;8989;48639; : : : (A001850 of The On-Line Encyclopedia of Integer Sequences) will be deflned so that dn counts the lattice paths running from (0;0) to (n; n) that use the steps (1;0), (0;1), and (1;1). In a recreational spirit we give a collection of 29 conflgurations that these numbers count.

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Objects counted by the central delannoy numbers

The Narayana distribution

Let $d(n)$ count the lattice paths from $(0,0)$ to $(n,n)$ using the steps (0,1), (1,0), and (1,1). Let $e(n)$ count the lattice paths from $(0,0)$ to $(n,n)$ with permitted steps from the step set ${\bf N} \times {\bf N} - \{(0,0)\}$, where ${\bf N}$ denotes the nonnegative integers. We give a bijective proof of the identity $e(n) = 2^{n-1} d(n)$ for $n \ge 1$. In giving perspective for our proof, we consider bijections between sets of lattice paths defined on various sets of permitted steps which yield path counts related to the Narayana polynomials.

/pdf/counting-lattice-paths-by-narayana-polynomials-41jelxr459.pdf

Counting Lattice Paths by Narayana Polynomials

http://www-igm.univ-mlv.fr/~fpsac/FPSAC06/SITE06/papers/66.pdf

Hankel determinants for some common lattice paths

Let ${\cal C}(d,n)$ denote the set of $d$-dimensional lattice paths using the steps $X_1 := (1, 0, \ldots, 0),$ $ X_2 := (0, 1, \ldots, 0),$ $\ldots,$ $ X_d := (0,0, \ldots,1)$, running from $(0,\ldots,0)$ to $(n,\ldots,n)$, and lying in $\{(x_1,x_2, \ldots, x_d) : 0 \le x_1 \le x_2 \le \ldots \le x_d \}$. On any path $P:=p_1p_2 \ldots p_{dn} \in {\cal C}(d,n)$, define the statistics ${\rm asc}(P) := $$|\{i : p_ip_{i+1} = X_jX_{\ell}, j \ell \}|$. Define the generalized Narayana number $N(d,n,k)$ to count the paths in ${\cal C}(d,n)$ with ${\rm asc}(P)=k$. We consider the derivation of a formula for $N(d,n,k)$, implicit in MacMahon's work. We examine other statistics for $N(d,n,k)$ and show that the statistics ${\rm asc}$ and ${\rm des}-d+1$ are equidistributed. We use Wegschaider's algorithm, extending Sister Celine's (Wilf-Zeilberger) method to multiple summation, to obtain recurrences for $N(3,n,k)$. We introduce the generalized large Schroder numbers $(2^{d-1}\sum_k N(d,n,k)2^k)_{n\ge1}$ to count constrained paths using step sets which include diagonal steps.

/pdf/generalizing-narayana-and-schroder-numbers-to-higher-51j65d0ud3.pdf

Robert A. Sulanke

Papers

Objects counted by the central delannoy numbers

The Narayana distribution

Counting Lattice Paths by Narayana Polynomials

Hankel determinants for some common lattice paths

Generalizing Narayana and Schroder Numbers to Higher Dimensions