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Rakesh Shivanna

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3 papers

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AAAI Conference 2019 Conference Paper

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

Aadirupa Saha
Rakesh Shivanna
Chiranjib Bhattacharyya

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 Ω(n2 ) 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 locality: Neighboring items are similar in their rankings. Noting the preferential nature of the data, we choose to embed not the graph, but, its 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—orthonormal representations—that includes (normalized) Laplacian as its special case. Our proposed algorithm, Pref-Rank, predicts the underlying ranking using an SVM based approach using the chosen embedding of the product graph, and is the first to provide statistical consistency on two ranking losses: Kendall’s tau and Spearman’s footrule, with a required sample complexity of O(n2 χ(Ḡ)) 2 3 pairs, χ(Ḡ) being the chromatic number of the complement graph Ḡ. Clearly, our sample complexity is smaller for dense graphs, with χ(Ḡ) characterizing the degree of node connectivity, which is also intuitive due to the locality assumption e. g. O(n 4 3 ) for union of k-cliques, or O(n 5 3 ) for random and power law graphs etc. —a quantity much smaller than the fundamental limit of Ω(n2 ) 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-world datasets, where our algorithm is shown to outperform the state of the art methods.

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NeurIPS Conference 2015 Conference Paper

Spectral Norm Regularization of Orthonormal Representations for Graph Transduction

Rakesh Shivanna
Bibaswan Chatterjee
Raman Sankaran
Chiranjib Bhattacharyya
Francis Bach

Recent literature~\cite{ando} 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 embeddings 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 Lov\'{a}sz~$\vartheta$ function. The main contribution of the paper is SPORE, a SPectral regularized ORthonormal Embedding for graph transduction, derived from the PAC bound. SPORE is posed as a non-smooth convex function over an \emph{elliptope}. These problems are usually solved as semi-definite programs (SDPs) with time complexity $O(n^6)$. We present, Infeasible Inexact proximal~(IIP): an Inexact proximal method which performs subgradient procedure on an approximate projection, not necessarily feasible. IIP is more scalable than SDP, has an $O(\frac{1}{\sqrt{T}})$ 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 $O(\frac{1}{\sqrt{T}})$. 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.

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NeurIPS Conference 2014 Conference Paper

Learning on graphs using Orthonormal Representation is Statistically Consistent

Rakesh Shivanna
Chiranjib Bhattacharyya

Existing research \cite{reg} suggests that embedding graphs on a unit sphere can be beneficial in learning labels on the vertices of a graph. However the choice of optimal embedding remains an open issue. \emph{Orthonormal representation} of graphs, a class of embeddings over the unit sphere, was introduced by Lov\'asz \cite{lovasz_shannon}. In this paper, we show that there exists orthonormal representations which are statistically consistent over a large class of graphs, including power law and random graphs. This result is achieved by extending the notion of consistency designed in the inductive setting to graph transduction. As part of the analysis, we explicitly derive relationships between the Rademacher complexity measure and structural properties of graphs, such as the chromatic number. We further show the fraction of vertices of a graph $G$, on $n$ nodes, that need to be labelled for the learning algorithm to be consistent, also known as labelled sample complexity, is $ \Omega\left(\frac{\vartheta(G)}{n}\right)^{\frac{1}{4}}$ where $\vartheta(G)$ is the famous Lov\'asz~$\vartheta$ function of the graph. This, for the first time, relates labelled sample complexity to graph connectivity properties, such as the density of graphs. In the multiview setting, whenever individual views are expressed by a graph, it is a well known heuristic that a convex combination of Laplacians \cite{lap_mv1} tend to improve accuracy. The analysis presented here easily extends to Multiple graph transduction, and helps develop a sound statistical understanding of the heuristic, previously unavailable.

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