Microsoft Excel is omni present. It is also an excellent vehicle for implementing many algorithms in their basic form to gain a better understanding of the underlying concepts. In my previous post, I had provided an Excel implementation of a classification problem to help understand naive Bayes classifier. In this post I want to demonstrate K-means clustering via an Excel implementation.Clustering implies partitioning data into meaningful groups such that data items within each group are similar but are dissimilar across different groups. One of the most widely used clustering procedure is the K-means clustering algorithm, which seeks to group data into K clusters. The K-means algorithm is meant for situations where data items are represented as vectors of continuous attributes, i.e. numeric attributes, and each cluster is represented by the mean/centroid of respective cluster members. The similarity between different data items in such a case is measured by the Euclidean distance. The steps of the K-means algorithm are:Step 1: Randomly assign every data item to one of the K clusters. (K is a user specified parameter)

This is my third post on tensors and tensor decompositions. The first post on this topic primarily introduced tensors and some related terminology. The second post was meant to illustrate a particularly simple tensor decomposition method, called the CP decomposition. In this post, I will describe another tensor decomposition method, known as the Tucker decomposition. While the CP decomposition’s chief merit is its simplicity, it is limited in its approximation capability and it requires the same number of components in each mode. The Tucker decomposition, on the other hand, is extremely efficient in terms of approximation and allows different number of components in different modes. Before going any further, lets look at factor matrices and n-mode product of a tensor and a matrix. Factor Matrices Recall the CP decomposition of an order three tensor expressed as X≈∑r=1Rar∘br∘cr, where (∘ ) represents the outer product. We can also represent this decomposition in terms of organizing the vectors, ar,br,cr,r=1,⋯R , into three matrices, A, B, and C, as A=[a1a2⋯aR], B=[b1b2⋯bR],and C=[c1c2⋯cR] The CP decomposition is then expressed as X≈[Λ;A,B,C], where Λ is a super-diagonal tensor with all zero entries except the diagonal elements. The matrices A, B, and C are called the factor matrices. Next, lets try to understand the n-mode product. Multiplying a Tensor and a Matrix How do you multiply a tensor and a matrix? The answer is via n-mode product. The n-mode product of a tensor X∈RI1×I2×⋯IN with a matrix U∈RJ1×In is a tensor of size I1×I2×⋯In−1×J×In+1×⋯×IN, and is denoted by X×nU . The product is calculated by multiplying each mode-n fibre by the U matrix. Lets look at an example to better understand the n-mode product. Lets consider a 2x2x3 tensor whose frontal slices are:

In Part 1 of this blog, I had explained what tensors are and some related terminology. Just to briefly recap, a tensor is a multidimensional array of numbers. Tensors provide a useful representation for data that involves multiple moda