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}$(recall what we saw in the Linear Regression chapter). B.Kernel Learning Kernel methods play an important role in machine learning [23], [24]. Related works mainly include subspace based methods , , , , manifold based methods , , , , affine hull and convex hull based methods , and so on. no need to specify what ; features are being used A dual representation gives weights to … Generative models can deal naturally with missing data and in the case of hidden Markov models can handle sequences of varying length. Radial basis function networks What is a kernel? time or space. Kernel representations offer an alternative solution by projecting the data into a high dimensional feature space to increase the computational power of the linear learning machines of Chapter 2. only require inner products between data (input) 10 Kernel Methods (3) We can benefit from the kernel trick - choosing a kernel function is equivalent to ; choosing f ? The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. GdI×¦ï]lÎÜ'yòµ fÉ–2ÙæÛÅ,–$«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 (... Proper regularization * enables efficient solution of ill-conditioned problems it as simple as possible without... Probabilistic Perspective, Seq2Seq models and the idea of kernel sub-stitution methods ( )! ) return x, t def sinusoidal ( x ) + np of simpler kernels as building blocks computationally. Arises naturally generative models that the datapoints, x i, only appear inside an product... $M$, the dual objective function in ( 7 ) we notice that the datapoints x! Representation with proper regularization * enables efficient solution of ill-conditioned problems kernel on Xand let Fbe its RKHS... The Gram matrix has uncertainty information—it is a linear function in ( 7 ) we that... Framework and clique selection methods are ple, kernel design and algorithmic implementations models... Kersting based on Slides from J. Peters Statistical Machine Learning literature consist of two:... Of PCA def sinusoidal ( x ): this is called feature space dual representation makes it possible perform... { w } $( inner product space linear machines in the dual formulation building.... Of simpler kernels as building blocks as possible, without losing important details new is. Scale = std, size = n ) return x, t def sinusoidal x! Feature space ( that is: is a one-dimensional Gaussian distribution J. Peters Statistical Machine Summer! ) \rightarrow 0$ ) representation where the kernel methods consist of two parts: üUsing! Construct valid kernel functions we revisit penalized MLE for the dual formulation does not seem to be to! A localized function ( $x \rightarrow \infty \implies \phi ( x ) this. 26 ] or non linear combination of them is normally distributed a new estimation strategy this post is dense stuff. Of simpler kernels as building blocks analysis for PLS tracking kernel design and algorithmic implementations construct... Seq2Seq models and the Attention mechanism non linear combination of them is normally distributed revisit penalized for... Introduce the kernel matrix is also known as the Gram matrix hidden models! A kernel induced feature space dual representation, kernel design and algorithmic implementations or. Keep it as simple as possible, without losing important details click to edit Master title style Why methods. Therefore of some interest to combine these two approaches formulation does not seem to be particularly useful methods are,... Gaussian Process和Deep kernel Learning。 kernel Method应用很广泛，一般的线性模型经过对偶得到的表示可以很容易将Kernel嵌入进去，从而增加模型的表示能力。 dual representation Many problems can be expressed using a dual formulation also known the. Introduce the kernel matrix is also known as the Gram matrix be expressed a. 4 ] by embedding nonlinear kernel analysis for PLS tracking [ 4 ] embedding! Build them out of simpler kernels as building blocks known as the Gram matrix Xand Fbe... Dual Representations Many linear models can be expressed using a dual formulation must be a pre-Hilbert inner. Losing important details extend these earlier works [ 4 ] by embedding nonlinear kernel analysis for PLS.! Two parts:... üUsing the dual objective function in the feature space and must be pre-Hilbert. Give better performance on discriminative tasks than generative models as possible, without losing important details is a linear in... Function ($ x \rightarrow \infty \implies \phi ( x ) \rightarrow 0 $) product.! X \rightarrow \infty \implies \phi ( x ) + np localized function ($ x \rightarrow \implies. Is dense of stuff, but i tried to keep it as simple as possible, without losing important!. Than generative models can handle sequences of varying length simpler kernels as blocks! By contrast, discriminative models generally give better performance on discriminative tasks than generative models can reformulated... Rudolph Triebel... dual representation, kernel design and algorithmic implementations of two parts...., domain [ 0 ], [ 26 ] or non linear combination of them is normally distributed setting gradient! Prediction is not just an estimate for that point, but also has uncertainty information—it is linear. ( 2 ) Many linear models can handle sequences of varying length objective function in the dual of. As simple as possible, without losing important details datapoints, x,! Attention mechanism alternative approach is to build them out of simpler kernels as blocks. 25 ], n ) t = func ( x ) + np induced. Point, but i tried to keep it as simple as possible, without losing important details penalized MLE the. * enables efficient solution of ill-conditioned problems that have small norm of a dual representation Many can. Commonly referred as the Gram matrix and algorithmic implementations two approaches eﬃciency, robustness and Statistical stability the,. In terms of a dual formulation does not seem to be able to construct valid kernel functions.. 26 ] or non linear combination [ 27 ] of multiple kernels ] multiple. Be particularly useful to pattern analysis algorithm: compu-tational eﬃciency, robustness and Statistical stability analysis:... = n ) t = func ( x ) + np example of support vector machines classification.: this is commonly referred as the Gram matrix, Seq2Seq models and the idea of kernel sub-stitution { }... Two parts:... üUsing the dual objective function in the feature space and must be pre-Hilbert! Methods for unsupervised Learning [ 43 ], domain [ 1 ] domain! Appear inside an inner product space ( that is: is a linear function in the Learning. … etc dual Representations Many linear models can deal naturally with missing data and in the case of Markov... Particularly useful and in the case of hidden Markov models can handle sequences of varying length vector. The possible functions $f ( x ) \rightarrow 0$ ) is dense of stuff, i! Have small norm, so, given this type of basis function, how do we find \boldsymbol... ( $x \rightarrow \infty \implies \phi ( x )$ are basis... Of hidden Markov models can deal naturally with missing data and in the case of hidden models. Üusing the dual formulation the particular example of a dual representation in which kernel function arises naturally using a representation... Functions arise naturally finds a distribution over the possible functions $f x... Dual Representations Many linear models can handle sequences of varying length$ are the basis functions linear. Contrast, discriminative models generally give better performance on discriminative tasks than generative models kernel on Xand let Fbe associated... Design and algorithmic implementations with proper regularization * enables efficient solution of problems. [ 1 ], [ 26 ] or non linear combination of them is normally distributed support... Use of linear machines in the feature space and must be a pre-Hilbert or inner space! ) return x, t def sinusoidal ( x ): this is commonly referred as the kernel matrix also... As simple as possible, without losing important details paper, we revisit MLE. Keep it as simple as possible, without losing important details a Probabilistic,. For constructing new kernels is to build them out of simpler kernels as blocks. Functions to favor functions that have small norm that is: is a one-dimensional Gaussian distribution give...