Algorithms Reference Matrix Factorization

Principal Component Analysis

PCA Description

Principal Component Analysis (PCA) is a simple, non-parametric method to transform the given data set with possibly correlated columns into a set of linearly uncorrelated or orthogonal columns, called principal components. The principal components are ordered in such a way that the first component accounts for the largest possible variance, followed by remaining principal components in the decreasing order of the amount of variance captured from the data. PCA is often used as a dimensionality reduction technique, where the original data is projected or rotated onto a low-dimensional space with basis vectors defined by top-$K$ (for a given value of $K$) principal components.

PCA Details

Principal Component Analysis (PCA) is a non-parametric procedure for orthogonal linear transformation of the input data to a new coordinate system, such that the greatest variance by some projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. In other words, PCA first selects a normalized direction in $m$-dimensional space ($m$ is the number of columns in the input data) along which the variance in input data is maximized – this is referred to as the first principal component. It then repeatedly finds other directions (principal components) in which the variance is maximized. At every step, PCA restricts the search for only those directions that are perpendicular to all previously selected directions. By doing so, PCA aims to reduce the redundancy among input variables. To understand the notion of redundancy, consider an extreme scenario with a data set comprising of two variables, where the first one denotes some quantity expressed in meters, and the other variable represents the same quantity but in inches. Both these variables evidently capture redundant information, and hence one of them can be removed. In a general scenario, keeping solely the linear combination of input variables would both express the data more concisely and reduce the number of variables. This is why PCA is often used as a dimensionality reduction technique.

The specific method to compute such a new coordinate system is as follows – compute a covariance matrix $C$ that measures the strength of correlation among all pairs of variables in the input data; factorize $C$ according to eigen decomposition to calculate its eigenvalues and eigenvectors; and finally, order eigenvectors in the decreasing order of their corresponding eigenvalue. The computed eigenvectors (also known as loadings) define the new coordinate system and the square root of eigen values provide the amount of variance in the input data explained by each coordinate or eigenvector.

PCA Returns

When MODEL is not provided, PCA procedure is applied on INPUT data to generate MODEL as well as the rotated data OUTPUT (if PROJDATA is set to $1$) in the new coordinate system. The produced model consists of basis vectors MODEL$/dominant.eigen.vectors$ for the new coordinate system; eigen values MODEL$/dominant.eigen.values$; and the standard deviation MODEL$/dominant.eigen.standard.deviations$ of principal components. When MODEL is provided, INPUT data is rotated according to the coordinate system defined by MODEL$/dominant.eigen.vectors$. The resulting data is stored at location OUTPUT.

Matrix Completion via Alternating Minimizations

MCAM Description

Low-rank matrix completion is an effective technique for statistical data analysis widely used in the data mining and machine learning applications. Matrix completion is a variant of low-rank matrix factorization with the goal of recovering a partially observed and potentially noisy matrix from a subset of its revealed entries. Perhaps the most popular applications in which matrix completion has been successfully applied is in the context of collaborative filtering in recommender systems. In this setting, the rows in the data matrix correspond to users, the columns to items such as movies, and entries to feedback provided by users for items. The goal is to predict missing entries of the rating matrix. This implementation uses the alternating least-squares (ALS) technique for solving large-scale matrix completion problems.

MCAM Details

Given an $m \times n$ input matrix $V$ and a rank parameter $r \ll \min{(m,n)}$, low-rank matrix factorization seeks to find an $m \times r$ matrix $L$ and an $r \times n$ matrix $R$ such that $V \approx LR$, i.e., we aim to approximate $V$ by the low-rank matrix $LR$. The quality of the approximation is determined by an application-dependent loss function $\mathcal{L}$. We aim at finding the loss-minimizing factor matrices, i.e.,

\[\begin{equation} (L^*, R^*) = \textrm{argmin}_{L,R}{\mathcal{L}(V,L,R)} \end{equation}\]

In the context of collaborative filtering in the recommender systems it is often the case that the input matrix $V$ contains several missing entries. Such entries are coded with the 0 value and the loss function is computed only based on the nonzero entries in $V$, i.e.,

\[%\label{eq:loss} \mathcal{L}=\sum_{(i,j)\in\Omega}l(V_{ij},L_{i*},R_{*j})\]

where $L_{i*}$ and $R_{*j}$, respectively, denote the $i$th row of $L$ and the $j$th column of $R$, $\Omega={\omega_1,\dots,\omega_N}$ denotes the training set containing the observed (nonzero) entries in $V$, and $l$ is some local loss function.

ALS is an optimization technique that can be used to solve quadratic problems. For matrix completion, the algorithm repeatedly keeps one of the unknown matrices ($L$ or $R$) fixed and optimizes the other one. In particular, ALS alternates between recomputing the rows of $L$ in one step and the columns of $R$ in the subsequent step. Our implementation of the ALS algorithm supports the loss functions summarized in Table 16 commonly used for matrix completion [ZhouWSP08].

Table 16

Popular loss functions supported by our ALS implementation; $N_{i*}$ and $N_{*j}$, respectively, denote the number of nonzero entries in row $i$ and column $j$ of $V$.

Loss Definition
$\mathcal{L}_\text{Nzsl}$ $\sum_{i,j:V_{ij}\neq 0} (V_{ij} - [LR]_{ij})^2$
$\mathcal{L}_\text{Nzsl+L2}$ $\mathcal{L}\text{Nzsl} + \lambda \Bigl( \sum{ik} L_{ik}^2 + \sum_{kj} R_{kj}^2 \Bigr)$
$\mathcal{L}_\text{Nzsl+wL2}$ $\mathcal{L}\text{Nzsl} + \lambda \Bigl(\sum{ik}N_{i} L_{ik}^2 + \sum_{kj}N_{j} R_{kj}^2 \Bigr)$

Note that the matrix completion problem as defined in (1) is a non-convex problem for all loss functions from Table 16. However, when fixing one of the matrices $L$ or $R$, we get a least-squares problem with a globally optimal solution. For example, for the case of \(\mathcal{L}_\text{Nzsl+wL2}\) we have the following closed form solutions

\[\begin{aligned} L^\top_{n+1,i*} &\leftarrow (R^{(i)}_n {[R^{(i)}_n]}^\top + \lambda N_2 I)^{-1} R_n V^\top_{i*}, \\ R_{n+1,*j} &\leftarrow ({[L^{(j)}_{n+1}]}^\top L^{(j)}_{n+1} + \lambda N_1 I)^{-1} L^\top_{n+1} V_{*j}, \end{aligned}\]

where $L_{n+1,i*}$ (resp. $R_{n+1,*j}$) denotes the $i$th row of $L_{n+1}$ (resp. $j$th column of $R_{n+1}$), $\lambda$ denotes the regularization parameter, $I$ is the identity matrix of appropriate dimensionality, $V_{i*}$ (resp. $V_{*j}$) denotes the revealed entries in row $i$ (column $j$), $R^{(i)}n$ ( resp. $L^{(j)}{n+1}$ ) refers to the corresponding columns of $R_n$ (rows of $L_{n+1}$), and $N_1$ (resp. $N_2$) denotes a diagonal matrix that contains the number of nonzero entries in row $i$ (column $j$) of $V$.

Prediction. Based on the factor matrices computed by ALS we provide two prediction scripts:

  1. ALS_predict.dml computes the predicted ratings for a given list of users and items.
  2. ALS_topk_predict.dml computes top-K item (where $K$ is given as input) with highest predicted ratings together with their corresponding ratings for a given list of users.

MCAM Returns

We output the factor matrices $L$ and $R$ after the algorithm has converged. The algorithm is declared as converged if one of the two criteria is meet: (1) the decrease in the value of loss function falls below thr given as an input parameter (if parameter check=TRUE), or (2) the maximum number of iterations (defined as parameter maxi) is reached. Note that for a given user $i$ prediction is possible only if user $i$ has rated at least one item, i.e., row $i$ in matrix $V$ has at least one nonzero entry. In case, some users have not rated any items the corresponding factor in $L$ will be all 0s. Similarly if some items have not been rated at all the corresponding factors in $R$ will contain only 0s. Our prediction scripts output the predicted ratings for a given list of users and items as well as the top-K items with highest predicted ratings together with the predicted ratings for a given list of users. Note that the predictions will only be provided for the users who have rated at least one item, i.e., the corresponding rows contain at least one nonzero entry.