Linear-time algorithms for the subpath kernel

The subpath kernel is a useful positive definite kernel, which takes arbitrary rooted trees as input, no matter whether they are ordered or unordered, We first show that the subpath kernel can exhibit excellent classification performance in combination with SVM through an intensive experiment. Secondly, we develop a theory of irreducible trees, and then, using it as a rigid mathematical basis, reconstruct a bottom-up linear-time algorithm for the subtree kernel, which is a correction of an algorithm well-known in the literature. Thirdly, we show a novel top-down algorithm, with which we can realize a linear-time parallel-computing algorithm to compute the subpath kernel

A subpath kernel for learning hierarchical image representations

International audienceTree kernels have demonstrated their ability to deal with hierarchical data, as the intrinsic tree structure often plays a discrimi-native role. While such kernels have been successfully applied to various domains such as nature language processing and bioinformatics, they mostly concentrate on ordered trees and whose nodes are described by symbolic data. Meanwhile, hierarchical representations have gained increasing interest to describe image content. This is particularly true in remote sensing, where such representations allow for revealing different objects of interest at various scales through a tree structure. However, the induced trees are unordered and the nodes are equipped with numerical features. In this paper, we propose a new structured kernel for hierarchical image representations which is built on the concept of subpath kernel. Experimental results on both artificial and remote sensing datasets show that the proposed kernel manages to deal with the hierarchical nature of the data, leading to better classification rates

The Weight Function in the Subtree Kernel is Decisive

Tree data are ubiquitous because they model a large variety of situations, e.g., the architecture of plants, the secondary structure of RNA, or the hierarchy of XML files. Nevertheless, the analysis of these non-Euclidean data is difficult per se. In this paper, we focus on the subtree kernel that is a convolution kernel for tree data introduced by Vishwanathan and Smola in the early 2000's. More precisely, we investigate the influence of the weight function from a theoretical perspective and in real data applications. We establish on a 2-classes stochastic model that the performance of the subtree kernel is improved when the weight of leaves vanishes, which motivates the definition of a new weight function, learned from the data and not fixed by the user as usually done. To this end, we define a unified framework for computing the subtree kernel from ordered or unordered trees, that is particularly suitable for tuning parameters. We show through eight real data classification problems the great efficiency of our approach, in particular for small datasets, which also states the high importance of the weight function. Finally, a visualization tool of the significant features is derived.Comment: 36 page

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted graph $G$ embedded on a surface of genus $g$ and a designated face $f$ bounded by a simple cycle of length $k$, uncovers a set $F \subseteq E(G)$ of size polynomial in $g$ and $k$ that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of $f$. We apply this general theorem to prove that: * given an unweighted graph $G$ embedded on a surface of genus $g$ and a terminal set $S \subseteq V(G)$, one can in polynomial time find a set $F \subseteq E(G)$ that contains an optimal Steiner tree $T$ for $S$ and that has size polynomial in $g$ and $|E(T)|$; * an analogous result holds for an optimal Steiner forest for a set $S$ of terminal pairs; * given an unweighted planar graph $G$ and a terminal set $S \subseteq V(G)$, one can in polynomial time find a set $F \subseteq E(G)$ that contains an optimal (edge) multiway cut $C$ separating $S$ and that has size polynomial in $|C|$. In the language of parameterized complexity, these results imply the first polynomial kernels for Steiner Tree and Steiner Forest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by the size of the cutset). Additionally, we obtain a weighted variant of our main contribution

Tree Echo State Networks

In this paper we present the Tree Echo State Network (TreeESN) model, generalizing the paradigm of Reservoir Computing to tree structured data. TreeESNs exploit an untrained generalized recursive reservoir, exhibiting extreme efficiency for learning in structured domains. In addition, we highlight through the paper other characteristics of the approach: First, we discuss the Markovian characterization of reservoir dynamics, extended to the case of tree domains, that is implied by the contractive setting of the TreeESN state transition function. Second, we study two types of state mapping functions to map the tree structured state of TreeESN into a fixed-size feature representation for classification or regression tasks. The critical role of the relation between the choice of the state mapping function and the Markovian characterization of the task is analyzed and experimentally investigated on both artificial and real-world tasks. Finally, experimental results on benchmark and real-world tasks show that the TreeESN approach, in spite of its efficiency, can achieve comparable results with state-of-the-art, although more complex, neural and kernel based models for tree structured data

Fast Computation of Shortest Smooth Paths and Uniformly Bounded Stretch with Lazy RPHAST

We study the shortest smooth path problem (SSPP), which is motivated by traffic-aware routing in road networks. The goal is to compute the fastest route according to the current traffic situation while avoiding undesired detours, such as briefly using a parking area to bypass a jammed highway. Detours are prevented by limiting the uniformly bounded stretch (UBS) with respect to a second weight function which disregards the traffic situation. The UBS is a path quality metric which measures the maximum relative length of detours on a path. In this paper, we settle the complexity of the SSPP and show that it is strongly NP-complete. We then present practical algorithms to solve the problem on continental-sized road networks both heuristically and exactly. A crucial building block of these algorithms is the UBS evaluation. We propose a novel algorithm to compute the UBS with only a few shortest path computations on typical paths. All our algorithms utilize Lazy RPHAST, a recently proposed technique to incrementally compute distances from many vertices towards a common target. An extensive evaluation shows that our algorithms outperform competing SSPP algorithms by up to two orders of magnitude and that our new UBS algorithm is the first to consistently compute exact UBS values in a matter of milliseconds

Efficient Hardware Acceleration of Robust Volumetric Light Transport Simulation

Efficiently simulating the full range of light effects in arbitrary input scenes that contain participating media is a difficult task. Unified points, beams and paths (UPBP) is an algorithm capable of capturing a wide range of media effects, by combining bidirectional path tracing (BPT) and photon density estimation (PDE) with multiple importance sampling (MIS). A computationally expensive task of UPBP is the MIS weight computation, performed each time a light path is formed. We derive an efficient algorithm to compute the MIS weights for UPBP, which improves over previous work by eliminating the need to iterate over the path vertices. We achieve this by maintaining recursive quantities as subpaths are generated, from which the subpath weights can be computed. In this way, the full path weight can be computed by only using the data cached at the two vertices at the ends of the subpaths. Furthermore, a costly part of PDE is the search for nearby photon points and beams. Previous work has shown that a spatial data structure for photon mapping can be implemented using the hardware-accelerated bounding volume hierarchy of NVIDIA's RTX GPUs. We show that the same technique can be applied to different types of volumetric PDE and compare the performance of these data structures with the state of the art. Finally, using our new algorithm and data structures we fully implement UPBP on the GPU which we, to the best of our knowledge, are the first to do so