More on piecewise linear approximation

Dynamic step selection algorithm for piecewise linear approximation of complex control trajectories

The object of research is the algorithm of piecewise linear approximation when applying it to the selection of facial features and compression of its images. One of the problem areas is to obtain the optimal ratio of the degree of compression and accuracy of image reproduction, as well as the accuracy of the obtained facial features, which can be used to search for people in databases. The main characteristics of the image of the face are the coordinates and size of the eyes, mouth, nose and other objects of attention. Dimensions, distances between them, as well as their relationship also form a set of characteristics. A piecewise linear approximation algorithm is used to identify and determine these features. First, it is used to approximate the image of the face to obtain a graph of the silhouette from right to left and, secondly, to approximate fragments of the face to obtain silhouettes of the face from top to bottom. The purpose of the next stage is to implement multilevel segmentation of the approximated images to cover them with rectangles of different intensity. Due to their shape they are called barcodes. These three stages of the algorithm the faces are represented by two barcode images are vertical and horizontal. This material is used to calculate facial features. The medium intensity function in a row or column is used to form an approximation object and as a tool to measure the values of facial image characteristics. Additionally, the widths of barcodes and the distances between them are calculated. Experimental results with faces from known databases are presented. A piecewise linear approximation is used to compress facial images. Experiments have shown how the accuracy of the approximation changes with the degree of compression of the image. The method has a linear complexity of the algorithm from the number of pixels in the image, which allows its testing for large data. Finding the coordinates of a synchronized object, such as the eyes, allows calculating all the distances between the objects of attention on the face in relative form. The developed software has control parameters for conducting research.

Solving function approximation problem is to appropriately find the relationship between dependent variable and independent variable(s). Function approximation algorithms normally require sufficient amount of samples to approximate a function. However, insufficient samples may result in unsatisfactory prediction to any function approximation algorithms. It is due to the failure of the function approximation algorithms to fill the information gap between the available and very limited samples. In this study, a function approximation algorithm which is based on Nadaraya-Watson Kernel Regression (NWKR) is proposed for approximating a non-linear function with small samples. Gaussian function is chosen as a kernel function for this study. The results show that the NWKR is effective in the case where the target function is non-linear and the given training sample is small. The performance of the NWKR is compared with other existing function approximation algorithms, such as artificial neural network.

Color machine vision for autonomous vehicles

We are concerned with computing bid prices in network revenue management using approximate linear programming. It is well-known that affine value function approximations yield bid prices which are not sensitive to remaining capacity. The analytic reduction to compact linear programs allows the efficient computation of such bid prices. On the other hand, capacity-dependent bid prices can be obtained using separable piecewise linear value function approximations. Even though compact linear programs have been derived for this case also, they are still computationally much more expensive compared to using affine functions. We propose compact linear programs requiring substantially smaller computing times while, simultaneously, significantly improving the performance of capacity-independent bid prices. This simplification is achieved by taking into account remaining capacity only if it becomes scarce. Although our proposed linear programs are relaxations of the unreduced approximate linear programs, we conjecture equivalence and provide according numerical support. We measure the quality of an approximation by the difference between the expected performance of an induced policy and the corresponding theoretical upper bound. Using this paradigm in numerical experiments, we demonstrate the competitiveness of our proposed linear programs.

In complex real-world environments, traditional (tabular) techniques for solving Reinforcement Learning (RL) do not scale. Function approximation is needed, but unfortunately, existing approaches generally have poor convergence and optimality guarantees. Additionally, for the case of human environments, it is valuable to be able to leverage human input. In this paper we introduce Expanding Value Function Approximation (EVFA), a function approximation algorithm that returns the optimal value function given sufficient rounds. To leverage human input, we introduce a new human-agent interaction scheme, training regimens, which allow humans to interact with and improve agent learning in the setting of a machine learning game. In experiments, we show EVFA compares favorably to standard value approximation approaches. We also show that training regimens enable humans to further improve EVFA performance. In our user study, we find that non-experts are able to provide effective regimens and that they found the game fun.

In reinforcement learning, when the state space is enormous or infinite, it is not feasible to find the exact value for each state in the memory. A common way to tackle this problem is to adopt linear value function approximation technique. In this paper, we review some commonly used linear value function approximation algorithms, which divided into two classes. The first class is the so-called on-policy algorithms that mainly include temporal difference and least-squares-based methods. The second is the off-policy algorithms that use two strategies to solve the exploration-exploitation problem. From the perspective of algorithm design, on-policy is simple, but it does not take both exploration and exploitation into consideration when collecting data, while the off-policy can well balance the relation between exploration and exploitation. In the end, we first present some experiments to demonstrate the usefulness of linear value function approximation technique in the continuous and high dimensional scenes, and then in-class and out-class experiments are respectively conducted to further illustrate the performance of different approximation algorithms.KeywordsReinforcement learningLinear value function approximationLeast-squares temporal difference

An algorithm for low-rank approximation of bivariate functions using splines

Residual gradient (RG) was proposed as an alternative to TD(0) for policy evaluation when function approximation is used, but there exists little formal analysis comparing them except in very limited cases. This paper employs techniques from online learning of linear functions and provides a worst-case (non-probabilistic) analysis to compare these two types of algorithms when linear function approximation is used. No statistical assumptions are made on the sequence of observations, so the analysis applies to non-Markovian and even adversarial domains as well. In particular, our results suggest that RG may result in smaller temporal differences, while TD(0) is more likely to yield smaller prediction errors. These phenomena can be observed even in two simple Markov chain examples that are non-adversarial.

The optimal minimax solution to the N segment piecewise linear approximation of arbitrary convex differentiable functions over a finite range is described. The optimal solution is uniquely described by the derivatives at N distinct points. The optimality of the solution is proven and a recursive descent algorithm is proposed. The efficacy of the algorithm and optimality of the solution are demonstrated in example solutions for commonly used functions.

A linearization method suitable for smart sensors and intelligent transducers with highly nonlinear transfer curve of sensors is proposed here. The first, abscissa, is transformed by one piece-wise linear approximation, and then the standard piece-wise approximation of attained function is performed. The method is compared with the standard piece-wise polynomial and linear approximation, based on the example of the negative temperature coefficient sensor.

Feedback particle filter (FPF) is a numerical algorithm to approximate the solution of the nonlinear filtering problem in continuous-time settings. In any numerical implementation of the FPF algorithm, the main challenge is to numerically approximate the so-called gain function. A numerical algorithm for gain function approximation is the subject of this paper. The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian $\Delta_\rho$. The numerical problem is to approximate this solution using only finitely many particles sampled from the probability distribution $\rho$. A diffusion map-based algorithm was proposed by the authors in prior works [A. Taghvaei and P. G. Mehta, Gain function approximation in the feedback particle filter, in 2016 IEEE 55th Conference on Decision and Control (CDC), IEEE, 2016, pp. 5446--5452], [A. Taghvaei, P. G. Mehta, and S. P. Meyn, Error estimates for the kernel gain function approximation in the feedback particle filter, in American Control Conference (ACC), IEEE, 2017, pp. 4576--4582] to solve this problem. The algorithm is named as such because it involves, as an intermediate step, a diffusion map approximation of the exact semigroup $e^{\Delta_\rho}$. The original contribution of this paper is to carry out a rigorous error analysis of the diffusion map-based algorithm. The error is shown to include two components: bias and variance. The bias results from the diffusion map approximation of the exact semigroup. The variance arises because of finite sample size. Scalings and upper bounds are derived for bias and variance. These bounds are then illustrated with numerical experiments that serve to emphasize the effects of problem dimension and sample size. The proposed algorithm is applied to two filtering examples and comparisons provided with the sequential importance resampling (SIR) particle filter.

On some aspects of approximation of ridge functions

We study the problem of computing correlation between two piecewise-linear bivariate functions defined over a common domain, where the surfaces they define in three dimensions---polyhedral terrains---can be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We present a randomized algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations of one of the terrains, in $O(n^{4/3}\\operatorname{polylog}n)$ expected time, where $n$ is the total number of vertices in the graphs of the two functions. We also present approximation algorithms for minimizing the mean distance between the graphs of univariate and bivariate functions. For univariate functions we present a $(1+\\varepsilon)$-approximation algorithm that runs in $O(n (1 + \\log^2 (1/\\varepsilon)))$ expected time for any fixed $\\varepsilon >0$. The $(1+\\varepsilon)$-approximation algorithm for bivariate functions runs in $O(n/\\varepsilon)$ time, for any fixed $\\varepsilon >0$, provided the two functions are defined over the same triangulation of their domain.

The goal of this paper is to investigate linear approximations of random functions and permutations. Our motivation is twofold. First, before the distinguishability of a practical cipher from an ideal one can be analysed, the cryptanalyst must have an accurate understanding of the statistical behaviour of the ideal cipher. Secondly, this issue has been neglected both in old and in more recent studies, particularly when multiple linear approximations are being used simultaneously. Traditional models have been based on the average behaviour and simplified using other assumptions such as independence of the linear approximations. Multidimensional cryptanalysis was introduced to avoid making artificial assumptions about statistical independence of linear approximations. On the other hand, it has the drawback of including many trivial approximations that do not contribute to the attack but just cause a waste of time and memory. We show for the first time in this paper that the trivial approximations reduce the degree of freedom of the related χ2 distribution. Previously, the affine linear cryptanalysis was proposed to allow removing trivial approximations and, at the same time, admitting a solid statistical model. In this paper, we identify another type of multidimensional linear approximation, called Davies-Meyer approximation, which has similar advantages, and present full statistical models for both the affine and the Davies-Meyer type of multidimensional linear approximations. The new models given in this paper are realistic, accurate and easy to use. They are backed up by standard statistical tools such as Pearson’s χ2 test and finite population correction and demonstrated to work accurately using practical examples.