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===== Part 1: SVM with no bias term ===== | ===== Part 1: SVM with no bias term ===== | ||
- | Formulate a soft-margin SVM without the bias term, i.e. $f(\x) = \w^{\tr} \x$. | + | Formulate a soft-margin SVM without the bias term, i.e. $f(\mathbf{x}) = \mathbf{w}^{T} \mathbf{x}$. |
Derive the saddle point conditions, KKT conditions and the dual. | Derive the saddle point conditions, KKT conditions and the dual. | ||
Compare it to the standard SVM formulation. | Compare it to the standard SVM formulation. | ||
Line 10: | Line 10: | ||
Hint: consider the difference in the constraints. | Hint: consider the difference in the constraints. | ||
- | Discuss the merit of the bias-less formulation as the dimensionality | + | ===== Part 2: Closest Centroid Algorithm ===== |
- | of the data (or the feature space) is varied. | + | |
- | When using this SVM formulation it may be useful to add a constant to the | + | Express the closest centroid algorithm in terms of kernels, i.e. determine how the coefficients $\alpha_i$ will be determined using a given labeled dataset. |
- | kernel matrix. Explain why this can be beneficial. | + | |
+ | ===== Part 3: Using SVMs ===== | ||
+ | |||
+ | The data for this question comes from a database called SCOP (structural | ||
+ | classification of proteins), which classifies proteins into classes | ||
+ | according to their structure (download it from {{assignments:scop_motif.data|here}}). | ||
+ | The data is a two-class classification | ||
+ | problem | ||
+ | of distinguishing a particular class of proteins from a selection of | ||
+ | examples sampled from the rest of the SCOP database. | ||
+ | I chose to represent the proteins in | ||
+ | terms of their motif composition. A sequence motif is a | ||
+ | pattern of nucleotides/amino acids that is conserved in evolution. | ||
+ | Motifs are usually associated with regions of the protein that are | ||
+ | important for its function, and are therefore useful in differentiating between classes of proteins. | ||
+ | A given protein will typically contain only a handful of motifs, and | ||
+ | so the data is very sparse. It is also very high dimensional, since | ||
+ | the number of conserved patterns in the space of all proteins is | ||
+ | large. | ||
+ | The data was constructed as part of the following analysis of detecting distant relationships between proteins: | ||
+ | |||
+ | * A. Ben-Hur and D. Brutlag. [[http://bioinformatics.oxfordjournals.org/content/19/suppl_1/i26.abstract | Remote homology detection: a motif based approach]]. In: Proceedings, eleventh international conference on intelligent systems for molecular biology. Bioinformatics 19(Suppl. 1): i26-i33, 2003. | ||
+ | |||
+ | Download the dataset associated with this assignment from the homework | ||
+ | page of the course. | ||
+ | In this assignment we will explore the dependence of classifier accuracy on | ||
+ | the kernel, kernel parameters, kernel normalization, and SVM parameter. | ||
+ | The use of the SVM class is discussed in the PyML [[http://pyml.sourceforge.net/tutorial.html#svms|tutorial]]. | ||
+ | |||
+ | By default a dataset is instantiated with a linear kernel attached to it. | ||
+ | To use a different kernel you need to attach a new kernel to the dataset: | ||
+ | <code python> | ||
+ | >>> from PyML import ker | ||
+ | >>> data.attachKernel(ker.Gaussian(gamma = 0.1)) | ||
+ | </code> | ||
+ | or | ||
+ | <code python> | ||
+ | >>> from PyML import her | ||
+ | >>> data.attachKernel(ker.Polynomial(degree = 3)) | ||
+ | </code> | ||
+ | In this question we will consider both the Gaussian and polynomial kernels: | ||
+ | $$ | ||
+ | K_{gaus}(\mathbf{x}, \mathbf{x'}) = \exp(-\gamma || \mathbf{x} - \mathbf{x}' ||^2) | ||
+ | $$ | ||
+ | $$ | ||
+ | K_{poly}(\mathbf{x}, \mathbf{x'}) = (1 + \mathbf{x}^T \mathbf{x}') ^{p} | ||
+ | $$ | ||
+ | Plot the accuracy of the classifier, measured using the success rate and the area under the ROC curve | ||
+ | as a function of both the ridge parameter of the classifier, and the free parameter | ||
+ | of the kernel function. | ||
+ | Show a couple of representative cross sections of this plot for a given value | ||
+ | of the ridge parameter, and for a given value of the kernel parameter. | ||
+ | Comment on the results. When exploring the values of a continuous | ||
+ | classifier/kernel parameter it is | ||
+ | useful use values that are distributed on an exponential grid, | ||
+ | i.e. something like 0.01, 0.1, 1, 10, 100 (note that the degree of the | ||
+ | polynomial kernel is not such a parameter). | ||
+ | |||
+ | |||
+ | For this type of sparse dataset it is useful to normalize the input features before | ||
+ | training and testing your classifier. | ||
+ | One way to do so is to divide each input example by its norm. This is | ||
+ | accomplished in PyML by: | ||
+ | <code python> | ||
+ | >>> data.normalize() | ||
+ | </code> | ||
+ | Compare the results under this normalization with what you obtain | ||
+ | without normalization. | ||
+ | |||
+ | You can visualize the whole kernel matrix associated with the data using the following commands: | ||
+ | <code python> | ||
+ | >>> from PyML import ker | ||
+ | >>> ker.showKernel(data) | ||
+ | </code> | ||
+ | Explain the structure that you are seeing in the plot (it is more | ||
+ | interesting when the data is normalized). |