# dual representation kernel methods

!or modifying the kernel matrix (as seen below)!Or training a generative model, then extract kernel as described before www.support-vector.net Second Property of SVMs: SVMs are Linear Learning Machines, that ! Kernel methods owe their name to the use of kernel functions, which enable them to operate in a high-dimensional, implicit feature space without ever computing the coordinates of the data in that space, but rather by simply computing the inner products between the images of all pairs of data in the feature space. * e.g. 2(x,x0) k(x,x0) = k. 1(x,x0)k. 2(x,x0) k(x,x0) = xTAx0. Outline 1.Kernel Methods for Regression 2.Gaussian Processes Regression wN¥³J7)ŞPóêõtyˆ”…$HÁ¡HÃÈæ\Ã1�dwš!X,›Ú´Â¨“ßssÖ¶ŠÓìöú¹qtµÉ"ØÚ]7^+½«Ä{sà²ÉiÖ¨O!üÔÙWv“ãà©„Xˆ;œC3¤p—]©1qR˜èPPnZÛÓ²Ak@»Œş9zŒi((ËèQtûùq)£Ã™â²Q¯K ë´ñtÓÕuM˜ªZèõu¸dèB‘œÃ bõ®³*3 Y~Gvv3†É¢íKGŠP²h6}JnçæôsB¨Q�',¹ÒòöÔ›Å¹Oc»ûu„¿÷ Substituting$\boldsymbol{w} = \Phi^T\boldsymbol{a}$into$L_{\boldsymbol{w}}$gives,$L_{\boldsymbol{w}} = \frac{1}{2}\boldsymbol{a}^T\Phi\Phi^T\Phi\Phi^T\boldsymbol{a} - \boldsymbol{a}^T\Phi\Phi^T\boldsymbol{t} + \frac{1}{2}\boldsymbol{t}^T\boldsymbol{t} + \frac{\lambda}{2}\boldsymbol{a}^t\Phi\Phi^T\boldsymbol{a}$, In terms of the Gram matrix, the sum-of-squares error function can be written as,$L_{\boldsymbol{a}} = \frac{1}{2}\boldsymbol{a}^TKK\boldsymbol{a} - \boldsymbol{a}^TK\boldsymbol{t} + \frac{1}{2}\boldsymbol{t}^T\boldsymbol{t} + \frac{\lambda}{2}\boldsymbol{a}^tK\boldsymbol{a}$,$\boldsymbol{a} = (K + \lambda\boldsymbol{I_N})^{-1}\boldsymbol{t}$, If we substitute this back into the linear regression model, we obtain the following prediction for a new input$\boldsymbol{x}$,$y(\boldsymbol{x}) = \boldsymbol{w}^T\phi(\boldsymbol{x}) = a^T\Phi\phi(\boldsymbol{x}) = \boldsymbol{k}(\boldsymbol{x})^T(K+\lambda\boldsymbol{I_N})^{-1}\boldsymbol{t}$. A GP assumes that$p(f(x_1),…,f(x_N))$is jointly Gaussian, with some mean$\mu(x)$and covariance$\sum (x)$given by$\sum_{ij} = k(x_i,x_j)$, where$k$is a positive definite kernel function.$k(\boldsymbol{x},\boldsymbol{x’}) = k(||\boldsymbol{x}-\boldsymbol{x’}||)$, called homogeneous kernels and also known as,$k(\boldsymbol{x},\boldsymbol{x’}) = ck_1(\boldsymbol{x},\boldsymbol{x’})$,$k(\boldsymbol{x},\boldsymbol{x’}) = f(\boldsymbol{x})k_1(\boldsymbol{x},\boldsymbol{x’})f(\boldsymbol{x})$. The key idea is that if$x_i$and$x_j$are deemed by the kernel to be similar, then we expect the output of the function at those points to be similar, too. There exist various form of kernels functions: Consider a linear regression model in which the parameters are obtained by minimizing the regularized sum-of-squares error function,$L_{\boldsymbol{w}} = \frac{1}{2}\sum_{n=1}^{N}(\boldsymbol{w}^T\phi(\boldsymbol{x_n})-t_n)^2 + \frac{\lambda}{2}\boldsymbol{w}^t\boldsymbol{w}$, What we want is to make$\boldsymbol{w}$and$\phi$disappear. It is therefore of some interest to combine these two approaches. The solution to the dual problem is: 10 J (w)= 1 2 wT T w wT t + 1 2 tT t + 2 wT w 2R¬ëáÿ©°�“.� �4qùÿD‰–×nÿŸÀ¬(høÿ”p×öÿ›Şşs¦ÿ÷(wNÿïW !Ûÿk ÚÚvÿZ!6±½»¶�¨-Şş?QÊ«ÏÀ§¾€èäZá Údu9h Ñi{ÿ ¶ë7¹ü¾EÿaKë»8#!.�ß^?Q97'Q. In this case, we must ensure that the function we choose is a valid kernel, in other words that it corresponds to a scalar product in some (perhaps infinite dimensional) feature space. For example, consider the kernel function$k(\boldsymbol{x},\boldsymbol{z}) = (\boldsymbol{x}^T\boldsymbol{z})^2$in two dimensional space:$k(\boldsymbol{x},\boldsymbol{z}) = (\boldsymbol{x}^T\boldsymbol{z})^2 = (x_1z_1+x_2z_2)^2 = x_1^2z_1^2 + 2x_1z_1x_2z_2 + x_2^2z_2^2 = (x_1^2,\sqrt{2}x_1x_2,x_2^2)(z_1^2,\sqrt{2}z_1z_2,z_2^2)^T = \phi(\boldsymbol{x})^T\phi(\boldsymbol{z})$. Note that$\Phi$is not a square matrix, so we have to compute the pseudo-inverse:$\boldsymbol{w} = (\Phi^T\Phi)^{-1}\Phi^T\boldsymbol{y}$(recall what we saw in the Linear Regression chapter). B.Kernel Learning Kernel methods play an important role in machine learning , . Related works mainly include subspace based methods , , , , manifold based methods , , , , affine hull and convex hull based methods , and so on. no need to specify what ; features are being used A dual representation gives weights to … Generative models can deal naturally with missing data and in the case of hidden Markov models can handle sequences of varying length. Radial basis function networks What is a kernel? time or space. Kernel representations offer an alternative solution by projecting the data into a high dimensional feature space to increase the computational power of the linear learning machines of Chapter 2. only require inner products between data (input) 10 Kernel Methods (3) We can benefit from the kernel trick - choosing a kernel function is equivalent to ; choosing f ? The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. GdI×¦ï]lÎÜ'yòµ fÉ–2ÙæÛÅ,–$«ãß-úŸG¾i* ¹t%mb/àEes¨ln.ìu For the dual objective function in (7) we notice that the datapoints, x i, only appear inside an inner product. This is clearly a valid kernel function and it says that two inputs $\boldsymbol{x}$ and $\boldsymbol{x’}$ are similar if they both have high probabilities. $k(\boldsymbol{x},\boldsymbol{x’}) = \boldsymbol{x}^TA\boldsymbol{x’}$, where $A$ is a symmetric positive semidefinite matrix. More precisely, taken from the textbook Machine Learning: A Probabilistic Perspective: A GP defines a prior over functions, which can be converted into a posterior over functions once we have seen some data. For example, Chen et al. Computing dot products First, in 2-d. Use a dual representation AND! Kernel Methods Kernel Methods: An Introduction An IntroductionI Many linear parametric models can be re-cast into an equivalent \dual representation" in which the predictions are based on linear combinations of a kernel function evaluated at the training data points. A machine-learning algorithm that involves a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. memory-based method. Kernel methods CSE 250B Deviations from linear separability Noise Find a separator that minimizes a convex loss function related ... 2 Compute w ( x) using the dual representation. In this post I will give you an introduction to Generative Adversarial Networks, explaining the reasons behind their architecture and how they are trained. Instead of solving the log-likelihood equation directly, as in existing MLE methods, we exploit a doubly dual embedding technique that leads to a novel saddle-point reformulation for the MLE (along with its conditional distribution generalization) in sec:dual_mle. A necessary and sufficient condition for a function $k(\boldsymbol{x},\boldsymbol{x’})$ to be a valid kernel is that the Gram matrix $K$ is positive semidefinite for all possible choices of the set ${\boldsymbol{x_n}}$. Lectures will introduce the kernel exponential family and propose a new estimation strategy the case of hidden Markov models deal. Support vector machines for classification one powerful technique for constructing new kernels is to construct valid kernel functions arise?. Appear inside an inner product ) … etc them is normally distributed vector machines for classification kernel induced feature (... 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