Minimax Path Research Articles

A Large Size Image Classification Method Based on Semi-supervised Learning

Background: Semi-supervised learning in the machine learning community has received widespread attention. Semi-supervised learning can use a small number of tagged samples and a large number of untagged samples for efficient learning. Methods: In 2014, Kim proposed a new semi-supervised learning method: the minimax label propagation (MMLP) method. This method reduces time complexity to O (n), with a smaller computation cost and stronger classification ability than traditional methods. However, classification results are not accurate in large-scale image classifications. Thus, in this paper, we propose a semisupervised image classification method, which is an MMLP-based algorithm. The main idea is threefold: (1) Improving connectivity of image pixels by pixel sampling to reduce the image size, at the same time, reduce the diversity of image characteristics; (2) Using a recall feature to improve the MMLP algorithm; (3) through classification mapping, gaining the classification of the original data from the classification of the data reduction. Results: In the end, our algorithm also gains a minimax path from untagged samples to tagged samples. The experimental results proved that this algorithm is applicable to semi-supervised learning on small-size and that it can also gain better classification results for large-size image at the same time. Conclusion: In our paper, considering the connectivity of the neighboring matrix and the diversity of the characteristics, we used meanshift clustering algorithm, next we will use fuzzy energy clustering on our algorithm. We will study the function of these paths.

An experimental analysis of least-cost path models on ordinal-scaled raster surfaces

ABSTRACT Selection of optimal paths or sequences of cells from a grid of cells is one of the most basic functions of raster-based geographic information systems. For this function to work, it is often assumed that the optimality of a path can be evaluated by the sum of the weighted lengths of all its segments – weighted, i.e. by the underlying cell values. The validity of this assumption must be questioned, however, if those values are measured on a scale that does not permit arithmetic operations. Through computational experiments with randomly generated artificial landscapes, this paper compares two models, minisum and minimax path models, which aggregate the values of the cells associated with a path using the sum function and the maximum function, respectively. Results suggest that the minisum path model is effective if the path search can be translated into the conventional least-cost path problem, which aims to find a path with the minimum cost-weighted length between two terminuses on a ratio-scaled raster cost surface. On the other hand, the minimax path model is found mathematically sounder if the cost values are measured on an ordinal scale and practically useful if the problem is concerned not with the minimization of cost but with the maximization of some desirable condition such as suitability.

Structure-oriented networks of shape collections

We introduce a co-analysis technique designed for correspondence inference within large shape collections. Such collections are naturally rich in variation, adding ambiguity to the notoriously difficult problem of correspondence computation. We leverage the robustness of correspondences between similar shapes to address the difficulties associated with this problem. In our approach, pairs of similar shapes are extracted from the collection, analyzed and matched in an efficient and reliable manner, culminating in the construction of a network of correspondences that connects the entire collection. The correspondence between any pair of shapes then amounts to a simple propagation along the minimax path between the two shapes in the network. At the heart of our approach is the introduction of a robust, structure-oriented shape matching method. Leveraging the idea of projective analysis, we partition 2D projections of a shape to obtain a set of 1D ordered regions, which are both simple and efficient to match. We lift the matched projections back to the 3D domain to obtain a pairwise shape correspondence. The emphasis given to structural compatibility is a central tool in estimating the reliability and completeness of a computed correspondence, uncovering any non-negligible semantic discrepancies that may exist between shapes. These detected differences are a deciding factor in the establishment of a network aiming to capture local similarities. We demonstrate that the combination of the presented observations into a co-analysis method allows us to establish reliable correspondences among shapes within large collections.

Fast Minimax Path-Based Joint Depth Interpolation

We propose a fast minimax path-based depth interpolation method. The algorithm computes for each target pixel varying contributions from reliable depth seeds, and weighted averaging is used to interpolate missing depths. Compared with state-of-the-art joint geodesic upsampling method which selects the $K$ nearest seeds to interpolate missing depths with $O(Kn)$ complexity, our method does not need to limit the number of seeds to $K$ and reduces the computational complexity to $O(n)$ . In addition, the minimax path chooses a path with the smallest maximum immediate pairwise pixel difference on it, so it tends to preserve sharp depth discontinuities better. In contrast to the results of previous depth upsampling algorithms, our approach can provide accurate depths with fewer artifacts.

WITHDRAWN: Joint depth map interpolation using minimax paths

We propose a minimax path-based depth interpolation method. The algorithm computes for each target pixel varying contributions from reliable depth seeds, and weighted averaging is used to interpolate missing depths. Compared with state-of-the-art joint geodesic upsampling method which selects the K nearest seeds to interpolate missing depths with O(Kn) complexity, our method does not need to limit the number of seeds to K and reduces the computational complexity to O(n). In addition, the minimax path chooses a path with the smallest maximum immediate pairwise pixel difference on it, so it tends to preserve sharp depth discontinuities better. In contrast to the results of previous depth upsampling algorithms, our approach provides accurate depths with fewer artifacts.

Walking on Minimax Paths for k-NN Search

Link-based dissimilarity measures, such as shortest path or Euclidean commute time distance, base their distance on paths between nodes of a weighted graph. These measures are known to be better suited to data manifold with nonconvex-shaped clusters, compared to Euclidean distance, so that k-nearest neighbor (NN) search is improved in such metric spaces. In this paper we present a new link-based dissimilarity measure based on minimax paths between nodes. Two main benefits of minimax path-based dissimilarity measure are: (1) only a subset of paths is considered to make it scalable, while Euclidean commute time distance considers all possible paths; (2) it better captures nonconvex-shaped cluster structure, compared to shortest path distance. We define the total cost assigned to a path between nodes as Lp norm of intermediate costs of edges involving the path, showing that minimax path emerges from our Lp norm over paths framework. We also define minimax distance as the intermediate cost of the longest edge on the minimax path, then present a greedy algorithm to compute k smallest minimax distances between a query and N data points in O(log N + k log k) time. Numerical experiments demonstrate that our minimax k-NN algorithm reduce the search time by several orders of magnitude, compared to existing methods, while the quality of k-NN search is significantly improved over Euclidean distance.

Connectionist network for dynamic programming problems

Dynamic programming is well known as a powerful modelling technique for dealing with the issue of making optimal decisions sequentially. Many practical problems, such as finding shortest paths in route planning, and multi-stage optimal control, can be formulated as special cases of the general sequential decision process. The paper proposes a connectionist network architecture, called the binary-relation inference network, which solves a special class of dynamic programming problems in the continuous time. They include the all-pair solutions for a family of closed semi-ring path problems, such as shortest paths, transitive closure, minimum spanning tree, and minimax path problems. The all-pair inference network specifies a basic and uniform computation of its individual units, which then collectively emerge towards a global optimal solution. The computational order in its discrete-time variants, either as synchronous or asynchronous networks, bears a close resemblance to the Floyd–Warshall algorithm and doubling algorithm. However, the continuous-time inference network offers a significant speed advantage if its non-sequential computation nature can be exploited. Simulation results of using analogue VLSI implementation of the inference network for solving shortest-path problems are promising.

On the minimax path in a network

On the minimax path in a network

Optimal Minimax Path of a Single Service Unit on a Network to Nonservice Destinations

In this paper we consider the problem of finding an optimal path of a single service unit that travels toward a “nonservice” destination. Two types of objective functions are examined. One objective function is the minimization of the maximum distance (or weighted distance) between the moving service unit and any demand (node) of the network. The second objective function is the minimization of the total time period that the distance (or weighted distance) between the moving service unit and any node exceeds a response time threshold λ. For these two objective functions, we present algorithms which can be calculated, respectively, in 0(n3) and 0(n3 log n) elementary operations.

The minimax path in a search for a circle in a plane

AbstractThe problem has been considered by Isbell of determining the path that minimizes the maximum distance along the path to a line in the same plane whose distance from the starting point of the search is known, while its direction is unknown. We consider in this article the analogous problem for the search for a circle of known radius, of known distance from a starting point in its plane. It is further shown that Isbell's solution is a limiting case of the problem posed as the radius of the circle tends to infinity. The problem appears to be of some practical significance, since it is equivalent to that of searching for an object a given distance away which will be spotted when we get sufficiently close—that is, within a specific radius.