How Many Pairwise Preferences Do We Need to Rank A Graph Consistently?

We consider the problem of optimal recovery of true ranking of $n$ items from a randomly chosen subset of their pairwise preferences. It is well known that without any further assumption, one requires a sample size of $\Omega(n^2)$ for the purpose. We analyze the problem with an additional structure of relational graph $G([n],E)$ over the $n$ items added with an assumption of \emph{locality}: Neighboring items are similar in their rankings. Noting the preferential nature of the data, we choose to embed not the graph, but, its \emph{strong product} to capture the pairwise node relationships. Furthermore, unlike existing literature that uses Laplacian embedding for graph based learning problems, we use a richer class of graph embeddings---\emph{orthonormal representations}---that includes (normalized) Laplacian as its special case. Our proposed algorithm, {\it Pref-Rank}, predicts the underlying ranking using an SVM based approach over the chosen embedding of the product graph, and is the first to provide \emph{statistical consistency} on two ranking losses: \emph{Kendall's tau} and \emph{Spearman's footrule}, with a required sample complexity of $O(n^2 \chi(\bar{G}))^{\frac{2}{3}}$ pairs, $\chi(\bar{G})$ being the \emph{chromatic number} of the complement graph $\bar{G}$. Clearly, our sample complexity is smaller for dense graphs, with $\chi(\bar G)$ characterizing the degree of node connectivity, which is also intuitive due to the locality assumption e.g. $O(n^\frac{4}{3})$ for union of $k$-cliques, or $O(n^\frac{5}{3})$ for random and power law graphs etc.---a quantity much smaller than the fundamental limit of $\Omega(n^2)$ for large $n$. This, for the first time, relates ranking complexity to structural properties of the graph. We also report experimental evaluations on different synthetic and real datasets, where our algorithm is shown to outperform the state-of-the-art methods.Comment: In Thirty-Third AAAI Conference on Artificial Intelligence, 201

Spectral Norm Regularization of Orthonormal Representations for Graph Transduction

International audienceRecent literature [1] suggests that embedding a graph on an unit sphere leads to better generalization for graph transduction. However, the choice of optimal embedding and an efficient algorithm to compute the same remains open. In this paper, we show that orthonormal representations, a class of unit-sphere graph em-beddings are PAC learnable. Existing PAC-based analysis do not apply as the VC dimension of the function class is infinite. We propose an alternative PAC-based bound, which do not depend on the VC dimension of the underlying function class, but is related to the famous Lovász ϑ function. The main contribution of the paper is SPORE, a SPectral regularized ORthonormal Embedding for graph trans-duction, derived from the PAC bound. SPORE is posed as a non-smooth convex function over an elliptope. These problems are usually solved as semi-definite programs (SDPs) with time complexity O(n^6). We present, Infeasible Inexact prox-imal (IIP): an Inexact proximal method which performs subgradient procedure on an approximate projection, not necessarily feasible. IIP is more scalable than SDP, has an O(1 √ T) convergence, and is generally applicable whenever a suitable approximate projection is available. We use IIP to compute SPORE where the approximate projection step is computed by FISTA, an accelerated gradient descent procedure. We show that the method has a convergence rate of O(1 √ T). The proposed algorithm easily scales to 1000's of vertices, while the standard SDP computation does not scale beyond few hundred vertices. Furthermore, the analysis presented here easily extends to the multiple graph setting

EXPLORING THE POTENTIAL OF VELVET BEAN, MUCUNA PRURIENS (L) SEED ON GROWTH AND GONADAL DEVELOPMENT OF MONO-SEX COMMON MOLY POECILIA SPHENOPS (VALENCIENNES, 1846)

Mucuna pruriens, a rich source of L-dihydroxyphenylalanine, commonly known as L-DOPA and a precursor to dopamine, holds potential as a natural nutritional supplement. This study aimed to delve into the impact of incorporating M. pruriens seed powder (MpSP) into the feed on growth parameters and gonadal development of mono-sex common molly (Poecilia sphenops). The fish population was divided into three experimental groups, such as G1, G2, and G3, and a control group (C), each comprising 20 individuals. Over 45 days, the experimental groups were nourished with a commercial diet bolstered by MpSP in different concentrations (5, 7 and 10g/kg of feed, respectively). In contrast, the control group was provided with a regular diet devoid of the supplement. At the end of the experiment, MpSP demonstrated significant modulation (p<0.05) of growth performance metrics, including specific growth rate (SGR), length gain rate (LGR), body mass gain (BMG), and feed conversion ratio (FCR). Impressively, even the lower concentration of MpSP (5g/kg diet) yielded substantial increments in sperm count (p <0.05) and gonadosomatic index (GSI). These findings were corroborated by histological changes that reflected enhanced testicular development, consistently outperforming the control group. These outcomes collectively suggest the potential of velvet bean seed powder as a feasible, natural, and costeffective dietary supplement for enhancing growth and testicular development in mono-sex P. sphenops