Large scale SVD

SVD is a popular approach for factorization of a matrix and has several applications in many domains. It proceeds by factorizing a given matrix A m x n as A = UΣV^T, where U is a m x m unitary matrix, i.e. U^TU=I, Σ is a m x n diagonal matrix consisting of the eigenvalues and V is a n x n unitary matrix. It is popularly used to find the rank k approximation of a given matrix. One of the main problems of using SVD for large matrices is the computation of the eigenvector decomposition. Lets see how the eigenvector decomposition can be carried out for large sparse matrices.

Power Iteration : One of the simplest methods of computation of the largest eigenvalue of a large sparse matrix is the power iteration. It however only computes a single largest eigenvalue and may converge very slowly. The algorithm is as follows:
  1. Start with a random vector b0
  2. b_{k+1} = Ab_k || Ab_k||
This algorithm converges to the largest eigenvalue under the assumption that there is a single largest eigenvalue and the starting vector b0 has a non-zero component in the direction of the eigenvector associated with the largest eigenvalue. The next eigenvector is computed by restricting the algorithm to the null space of the known eigenvectors.

Arnoldi's algorithm :
Lanczos method :

Comments

Post a Comment

Popular posts from this blog

MinHash, Bloom Filters and LSH

Lets talk about some large scale algorithms widely used in the document world. MinHash: Minhash is a really cute algorithm to determine how does a document compare to another. One simple way to compute document similarity is to compute the jaccard similarity between them. The jaccard similarity between the documents is computed as follows: def jaccard_similarity(set1: set, set2: set): if len(set1) == 0 or len(set2) == 0: return 0.0 common = len(set1.intersection(set2)) if common == 0: return 0.0 union = len(set1.union(set2)) return common / union One problem with Jaccard similarity is that it can take a lot of time to compute especially for large scale documents. The idea behind minhashing is to first operate on the space of document shingles. Shingles are basically any combination of k consecutive words of the document. For example, for the sentence "hi i am jaya" the possible 3 shingles are "hi i am" and "i am jaya...
Read more

Perceptron Algorithm

Image
The perceptron algorithm is a linear classifier using a linear function (w'x + b) to classify the points into two classes. The loss function for the perceptron algorithm just counts the points misclassified and then incrementally adjusts the weight vector and thus the decision boundary to appropriately classify the two classes. Perhaps this is the simplest algorithm one can think of to update the weight vector and is shown as follows: w(k+1) = w(k) + yt*xt if yt ≠ f(xt; w) i.e. we cycle through the training points and adjust the weight vector only if the points are misclassified. Note that our class labels are -1 and 1 and hence if the points are misclassified then y_t* f(xt; w) will be negative. In order to understand why the seemingly simple update rule works and leads to a convergence, lets closely examine it. One of the underlying assumption in the perceptron model is that the points are linearly separated hence there exists an w* such that y_t * f(xt; w*) ≥ γ for all...
Read more

Logistic Regression

The logistic regression method even though called regression has a unique ability to give posterior probability values for a data point to belong to a particular class and hence serve as a discriminative function useful for classification. The idea is simple, to use the logistic sigmoid function on the linear function of feature vector. p(y = 1| x) = σ(w'x +b) The likelihood function can be written as p(t|w) = Π y_i^t_i*(1 - y_i)^(1 - t_i) ln p(t|w) = Σ t_i*ln(y_i) + (1 - t_i)*ln(1 - y_i) The error function is computed by taking the negative of the log likelihood function and is computed as follows: E(w) = -ln p(t|w) Now, ∇E(w) is computed as follows: y = 1 / (1 + exp(-w'x) Taking derivative of both sides, ∇y/∇w = exp(-w'x)(x)/(1 + exp(-w'x))^2 ∇y/∇w = xy(1-y) Substituting the value in ∇E(w), we get ∇ E(w) = Σ (y_i - t_i)*x_i The derivative of the error function is similar to the least squares problem, although w...
Read more