LowRankModels.jl is a julia package for modeling and fitting generalized low rank models (GLRMs). GLRMs model a data array by a low rank matrix, and include many well known models in data analysis, such as principal components analysis (PCA), matrix completion, robust PCA, nonnegative matrix factorization, k-means, and many more.

For more information on GLRMs, see our paper.

LowRankModels.jl makes it easy to mix and match loss functions and regularizers to construct a model suitable for a particular data set. In particular, it supports

• using different loss functions for different columns of the data array, which is useful when data types are heterogeneous (eg, real, boolean, and ordinal columns);
• fitting the model to only some of the entries in the table, which is useful for data tables with many missing (unobserved) entries; and
• adding offsets and scalings to the model without destroying sparsity, which is useful when the data is poorly scaled.

To install, just call

Pkg.clone("https://github.com/madeleineudell/LowRankModels.jl.git")

at the julia prompt.

GLRMs form a low rank model for tabular data A with m rows and n columns, which can be input as an array or any array-like object (for example, a data frame). It is fine if only some of the entries have been observed (i.e., the others are missing or NA); the GLRM will only be fit on the observed entries obs. The desired model is specified by choosing a rank k for the model, an array of loss functions losses, and two regularizers, rx and ry. The data is modeled as XY, where X is a mxk matrix and Y is a kxn matrix. X and Y are found by solving the optimization problem

minimize sum_{(i,j) in obs} losses[j](x[i,:] y[:,j], A[i,j]) + sum_i rx(x[i,:]) + sum_j ry(y[:,j])

The basic type used by LowRankModels.jl is the GLRM. To form a GLRM, the user specifies

• the data A
• the array of loss functions losses
• the regularizers rx and ry
• the rank k

The user may also specify

• the observed entries obs
• starting matrices X₀ and Y₀

obs is a list of tuples of the indices of the observed entries in the matrix, and may be omitted if all the entries in the matrix have been observed. X₀ and Y₀ are initialization matrices that represent a starting guess for the optimization.

Losses and regularizers must be of type Loss and Regularizer, respectively, and may be chosen from a list of supported losses and regularizers, which include

Losses:

• quadratic loss quadratic
• hinge loss hinge
• weighted hinge loss weighted_hinge
• l1 loss l1
• ordinal hinge loss ordinal_hinge
• periodic loss periodic

Regularizers:

• quadratic regularization quadreg
• l1 regularization onereg
• no regularization zeroreg
• nonnegative constraint nonnegative (eg, for nonnegative matrix factorization)
• 1-sparse constraint onesparse (eg, for orthogonal NNMF)
• unit 1-sparse constraint unitonesparse (eg, for k-means)

Users may also implement their own losses and regularizers; see loss_and_reg.jl for more details.

For example, the following code forms a k-means model with k=5 on the 100x100 matrix A:

using LowRankModels
m,n,k = 100,100,5
losses = fill(quadratic(),n)
rx = unitonesparse() # each row is assigned to exactly one cluster
ry = zeroreg() # no regularization on the cluster centroids
glrm = GLRM(A,losses,rt,r,k)

For more examples, see examples/simple_glrms.jl.

To fit the model, call

X,Y,ch = fit!(glrm)

which runs an alternating directions proximal gradient method on glrm to find the X and Y minimizing the objective function. (ch gives the convergence history; see Technical details below for more information.)

If not all entries are present in your data table, just tell the GLRM which observations to fit the model to by listing tuples of their indices in obs. Then initialize the model using

GLRM(A,losses,rx,ry,k, obs=obs)

If A is a DataFrame and you just want the model to ignore any entry that is of type NA, you can use

obs = observations(A)

By default, LowRankModels.jl adds proper offsets to your model scales the loss functions and regularizers so all columns have the same pull in the model. (For more about what these functions do, see the code or the paper.) To change this behavior, you can call glrm = GLRM(A,losses,rx,ry,k, offset=false, scale=false).

If you change your mind after creating the GLRM, you can also add offsets and scalings to previously unscaled models:

• Add an offset to the model (by applying no regularization to the last row of the matrix Y, and enforcing that the last column of X be all 1s) using

  add_offset!(glrm)
  

• Scale the loss functions and regularizers by calling

  equilibrate_variance!(glrm)
  

Perhaps all this sounds like too much work. Perhaps you happen to have a DataFrame df lying around that you'd like a low rank (eg, k=2) model for. For example,

using RDatasets
df = RDatasets.dataset("psych", "msq")

Never fear! Just call

glrm, labels = GLRM(df,2)
X, Y, ch = fit!(glrm)

This will fit a GLRM to your data, using a quadratic loss for real valued columns, hinge loss for boolean columns, and ordinal hinge loss for integer columns. (Right now, all other data types are ignored, as are NAs.) It returns the column labels for the columns it fit, along with the model.

You can use the model to get some intuition for the data set. For example, try plotting the columns of Y with the labels; you might see that similar features are close to each other!

The function fit! uses an alternating directions proximal gradient method to minimize the objective. This method is not guaranteed to converge to the optimum, or even to a local minimum. If your code is not converging or is converging to a model you dislike, there are a number of parameters you can tweak.

The algorithm starts with glrm.X and glrm.Y as the initial estimates for X and Y. If these are not given explicitly, they will be initialized randomly. If you have a good guess for a model, try setting them explicitly. If you think that you're getting stuck in a local minimum, try reinitializing your GLRM (so as to construct a new initial random point) and see if the model you obtain improves.

The function fit! sets the fields glrm.X and glrm.Y after fitting the model. This is particularly useful if you want to use the model you generate as a warm start for further iterations. If you prefer to preserve the original glrm.X and glrm.Y (eg, for cross validation), you should call the function fit, which does not mutate its arguments.

You can even start with an easy-to-optimize loss function, run fit!, change the loss function (glrm.losses = newlosses), and keep going from your warm start by calling fit! again to fit the new loss functions.

If you don't have a good guess at a warm start for your model, you might try one of the initializations provided in LowRankModels.

• init_svd! initializes the model as the truncated SVD of the matrix of observed entries, with unobserved entries filled in with zeros. This initialization is known to result in provably good solutions for a number of "PCA-like" problems. See our paper for details.
• init_kmeanspp! initializes the model using a modification of the kmeans++ algorithm for data sets with missing entries; see our paper for details. This works well for fitting clustering models, and may help in achieving better fits for nonnegative matrix factorization problems as well.

Parameters are encoded in a Parameter type, which sets the step size stepsize, number of rounds max_iter of alternating proximal gradient, and the convergence tolerance convergence_tol.

• The step size controls the speed of convergence. Small step sizes will slow convergence, while large ones will cause divergence. stepsize should be of order 1; autoencode scales it by the maximum number of entries per column or row so that step lengths remain of order 1.
• The algorithm stops when the decrease in the objective per iteration is less than convergence_tol*length(obs),
• or when the maximum number of rounds max_iter has been reached.

By default, the parameters are set to use a step size of 1, a maximum of 100 iterations, and a convergence tolerance of .001:

Params(1,100,.001)

ch gives the convergence history so that the success of the optimization can be monitored; ch.objective stores the objective values, and ch.times captures the times these objective values were achieved. Try plotting this to see if you just need to increase max_iter to converge to a better model.

A number of useful functions are available to help you check whether a given low rank model overfits to the test data set. These functions should help you choose adequate regularization for your model.

• cross_validate(glrm::GLRM, nfolds=5, params=Params(); verbose=false, use_folds=None): performs n-fold cross validation and returns average loss among all folds. More specifically, splits observations in glrm into nfolds groups, and builds use_folds new GLRMs, each with one group of observations left out. (use_folds defaults to nfolds.) Trains each GLRM and returns the average loss.
• cv_by_iter(glrm::GLRM, holdout_proportion=.1, params=Params(1,1,.01,.01), niters=30; verbose=true): computes the test error and train error of the GLRM as it is trained. Splits the observations into a training set (1-holdout_proportion of the original observations) and a test set (holdout_proportion of the original observations). Performs params.maxiter iterations of the fitting algorithm on the training set niters times, and returns the test and train error as a function of iteration.

• regularization_path(glrm::GLRM; params=Params(), reg_params=logspace(2,-2,5), holdout_proportion=.1, verbose=true, ch::ConvergenceHistory=ConvergenceHistory("reg_path")): computes the train and test error for GLRMs varying the scaling of the regularization through any scaling factor in the array reg_params.

• get_train_and_test(obs, m, n, holdout_proportion=.1): splits observations obs into a train and test set. m and n must be at least as large as the maximal value of the first or second elements of the tuples in observations, respectively. Returns observed_features and observed_examples for both train and test sets.

If you use LowRankModels for published work, we encourage you to cite the software.

Use the following BibTeX citation:

@article{udell2014,
    title = {Generalized Low Rank Models},
    author ={Udell, Madeleine and Horn, Corinne and Zadeh, Reza and Boyd, Stephen},
    year = {2014},
    archivePrefix = "arXiv",
    eprint = {1410.0342},
    primaryClass = "stat-ml",
    journal={arXiv preprint arXiv:1410.0342},
}