Chromatic PAC-Bayes Bounds for Non-IID Data: Applications to Ranking and Stationary β\beta-Mixing Processes

Pac-Bayes bounds are among the most accurate generalization bounds for classifiers learned from independently and identically distributed (IID) data, and it is particularly so for margin classifiers: there have been recent contributions showing how practical these bounds can be either to perform model selection (Ambroladze et al., 2007) or even to directly guide the learning of linear classifiers (Germain et al., 2009). However, there are many practical situations where the training data show some dependencies and where the traditional IID assumption does not hold. Stating generalization bounds for such frameworks is therefore of the utmost interest, both from theoretical and practical standpoints. In this work, we propose the first - to the best of our knowledge - Pac-Bayes generalization bounds for classifiers trained on data exhibiting interdependencies. The approach undertaken to establish our results is based on the decomposition of a so-called dependency graph that encodes the dependencies within the data, in sets of independent data, thanks to graph fractional covers. Our bounds are very general, since being able to find an upper bound on the fractional chromatic number of the dependency graph is sufficient to get new Pac-Bayes bounds for specific settings. We show how our results can be used to derive bounds for ranking statistics (such as Auc) and classifiers trained on data distributed according to a stationary {\ss}-mixing process. In the way, we show how our approach seemlessly allows us to deal with U-processes. As a side note, we also provide a Pac-Bayes generalization bound for classifiers learned on data from stationary $\varphi$-mixing distributions.Comment: Long version of the AISTATS 09 paper: http://jmlr.csail.mit.edu/proceedings/papers/v5/ralaivola09a/ralaivola09a.pd

Hopf bifurcation with non-semisimple 1:1 resonance

A generalised Hopf bifurcation, corresponding to non-semisimple double imaginary eigenvalues (case of 1:1 resonance), is analysed using a normal form approach. This bifurcation has linear codimension-3, and a centre subspace of dimension 4. The four-dimensional normal form is reduced to a three-dimensional system, which is normal to the group orbits of a phase-shift symmetry. There may exist 0, 1 or 2 small-amplitude periodic solutions. Invariant 2-tori of quasiperiodic solutions bifurcate from these periodic solutions. The authors locate one-dimensional varieties in the parameter space 1223 on which the system has four different codimension-2 singularities: a Bogdanov-Takens bifurcation a 1322 symmetric cusp, a Hopf/Hopf mode interaction without strong resonance, and a steady-state/Hopf mode interaction with eigenvalues (0, i,-i)

Incongruent restricted disjoint covering systems

We define an incongruent restricted disjoint covering system on [1,n] as a set of congruence classes such that each integer in the interval [1,n] belongs to exactly one class, and each class contains at least two members of the interval. In this paper we report some computational and structural results and present some open problems concerning such systems.7 page(s

A generalization of the problem of Mariusz Meszka

Mariusz Meszka has conjectured that given a prime p=2n+1 and a list L containing n positive integers not exceeding n there exists a near 1-factor in K_p whose list of edge-lengths is L. In this paper we propose a generalization of this problem to the case in which p is an odd integer not necessarily prime. In particular, we give a necessary condition for the existence of such a near 1-factor for any odd integer p. We show that this condition is also sufficient for any list L whose underlying set S has size 1, 2, or n. Then we prove that the conjecture is true if S={1,2,t} for any positive integer t not coprime with the order p of the complete graph. Also, we give partial results when t and p are coprime. Finally, we present a complete solution for t<12.Comment: 15 page

Starter sequences: generalizations and applications

In this thesis we introduce new types of starter sequences, pseudo-starter sequences, starter-labellings, and generalized (extended) starter sequences. We apply these new sequences to graph labeling. All the necessary conditions for the existence of starter, pseudo-starter, extended, m-fold, excess, and generalized (extended) starter sequences are determined, and some of these conditions are shown to be sufficient. The relationship between starter sequences and graph labellings is introduced. Moreover, the starter-labeling and the minimum hooked starter-labeling of paths, cycles, and k- windmills are investigated. We show that all paths, cycles, and k-windmills can be starter-labelled or minimum starter-labelled

BOOL-AN: A method for comparative sequence analysis and phylogenetic reconstruction

A novel discrete mathematical approach is proposed as an additional tool for molecular systematics which does not require prior statistical assumptions concerning the evolutionary process. The method is based on algorithms generating mathematical representations directly from DNA/RNA or protein sequences, followed by the output of numerical (scalar or vector) and visual characteristics (graphs). The binary encoded sequence information is transformed into a compact analytical form, called the Iterative Canonical Form (or ICF) of Boolean functions, which can then be used as a generalized molecular descriptor. The method provides raw vector data for calculating different distance matrices, which in turn can be analyzed by neighbor-joining or UPGMA to derive a phylogenetic tree, or by principal coordinates analysis to get an ordination scattergram. The new method and the associated software for inferring phylogenetic trees are called the Boolean analysis or BOOL-AN

Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

It is conjectured that for every pair $(\ell,m)$ of odd integers greater than 2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of $K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the others of type $m^{\ell}$. The authors prove the conjecture in the affirmative when $\ell \equiv 1\; \pmod{4}$ and $m \geq \ell^2 -\ell + 1$.Comment: 31 page