LowRankModels.jl
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, kmeans, and many more.
For more information on GLRMs, see our paper. There is a python interface to this package, and a GLRM implementation in the H2O machine learning platform with interfaces in a variety of languages.
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.
Installation
To install, just call
Pkg.add("LowRankModels")
at the julia prompt.
Generalized Low Rank Models
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 arraylike object (for example, a data frame).
It is fine if only some of the entries have been observed
(i.e., the others are missing
); the GLRM will only be fit on the !ismissing
entries.
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 X'*Y
, where X
is a k
xm
matrix and Y
is a k
xn
matrix.
X
and Y
are found by solving the optimization problem
minimize sum_{(i,j) in obs} losses[j]((X'*Y)[i,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
(anyAbstractArray
, such as an array, a sparse matrix, or a data frame)  the array of loss functions
losses
 the regularizers
rx
andry
 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.
If A
is a sparse matrix, implicit zeros are interpreted
as missing entries by default;
see the discussion of sparse matrices below for more details.
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
QuadLoss
 hinge loss
HingeLoss
 logistic loss
LogisticLoss
 poisson loss
PoissonLoss
 weighted hinge loss
WeightedHingeLoss
 l1 loss
L1Loss
 ordinal hinge loss
OrdinalHingeLoss
 periodic loss
PeriodicLoss
 multinomial categorical loss
MultinomialLoss
 multinomial ordinal (aka ordered logit) loss
OrderedMultinomialLoss
 biggervssmaller loss
BvSLoss
(for ordinal data)  onevs allloss
OvALoss
(for categorical data)
The constructors for all the ordinal and categorical losses take as an argument the maximum (or both minimum and maximum) value the variable may take. Using the onevsall loss is equivalent to transforming a categorical value to a onehot vector and using a binary loss on each entry in that vector. Using the biggervssmaller loss is equivalent to transforming the ordinal value to a boolean vector and using a binary loss on each entry in that vector. By default, the binary loss used is the logistic loss.
Regularizers:
 quadratic regularization
QuadReg
 constrained squared euclidean norm
QuadConstraint
 l1 regularization
OneReg
 no regularization
ZeroReg
 nonnegative constraint
NonNegConstraint
(eg, for nonnegative matrix factorization)  1sparse constraint
OneSparseConstraint
(eg, for orthogonal NNMF)  unit 1sparse constraint
UnitOneSparseConstraint
(eg, for kmeans)  simplex constraint
SimplexConstraint
 l1 regularization, combined with nonnegative constraint
NonNegOneReg
 fix features at values
y0
FixedLatentFeaturesConstraint(y0)
Each of these losses and regularizers can be scaled
(for example, to increase the importance of the loss relative to the regularizer)
by calling mul!(loss, newscale)
.
Users may also implement their own losses and regularizers,
or adjust internal parameters of the losses and regularizers;
see losses.jl and regularizers.jl for more details.
Example
For example, the following code forms a kmeans model with k=5
on the 100
x100
matrix A
:
using LowRankModels
m,n,k = 100,100,5
losses = QuadLoss() # minimize squared distance to cluster centroids
rx = UnitOneSparseConstraint() # each row is assigned to exactly one cluster
ry = ZeroReg() # no regularization on the cluster centroids
glrm = GLRM(A,losses,rx,ry,k)
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.)
The losses
argument can also be an array of loss functions,
with one for each column (in order). For example,
for a data set with 3 columns, you could use
losses = Loss[QuadLoss(), LogisticLoss(), HingeLoss()]
Similiarly, the ry
argument can be an array of regularizers,
with one for each column (in order). For example,
for a data set with 3 columns, you could use
ry = Regularizer[QuadReg(1), QuadReg(10), FixedLatentFeaturesConstraint([1.,2.,3.])]
This regularizes the first to columns of Y
with Y[:,1]^2 + 10Y[:,2]^2
and constrains the third (and last) column of Y
to be equal to [1,2,3]
.
Missing data
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
,
eg, if obs=[(1,2),(5,3)]
, exactly two entries have been observed.
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 missing
, you can use
obs = observations(A)
Standard low rank models
Low rank models can easily be used to fit standard models such as PCA, kmeans, and nonnegative matrix factorization. The following functions are available:
pca
: principal components analysisqpca
: quadratically regularized principal components analysisrpca
: robust principal components analysisnnmf
: nonnegative matrix factorizationkmeans
: kmeans
See the code for usage.
Any keyword argument valid for a GLRM
object,
such as an initial value for X
or Y
or a list of observations,
can also be used with these standard low rank models.
Scaling and offsets
If you choose, LowRankModels.jl can add an offset to your model and scale the loss functions and regularizers so all columns have the same pull in the model. Simply call
glrm = GLRM(A,losses,rx,ry,k, offset=true, scale=true)
This transformation generalizes standardization, a common proprocessing technique applied before PCA. (For more about offsets and scaling, see the code or the paper.)
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 ofX
be all 1s) usingadd_offset!(glrm)

Scale the loss functions and regularizers by calling
equilibrate_variance!(glrm)

Scale only the columns associated to
QuadLoss
orHuberLoss
loss functions.prob_scale!(glrm)
Fitting DataFrames
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,
import RDatasets
df = RDatasets.dataset("psych", "msq")
Never fear! Just call
glrm, labels = GLRM(df, k)
X, Y, ch = fit!(glrm)
This will fit a GLRM with rank k
to your data,
using a QuadLoss loss for real valued columns,
HingeLoss loss for boolean columns,
and ordinal HingeLoss loss for integer columns,
a small amount of QuadLoss regularization,
and scaling and adding an offset to the model as described here.
It returns the column labels for the columns it fit, along with the model.
Right now, all other data types are ignored.
NaN
values are treated as missing values (NA
s) and ignored in the fit.
The full call signature is
function GLRM(df::DataFrame, k::Int;
losses = Loss[], rx = QuadReg(.01), ry = QuadReg(.01),
offset = true, scale = false,
prob_scale = true, NaNs_to_NAs = true)
You can modify the losses or regularizers, or turn off offsets or scaling, using these keyword arguments.
Or to specify a map from data types to losses, define a new loss_map from datatypes to losses (like probabilistic_losses, below):
probabilistic_losses = Dict{Symbol, Any}(
:real => QuadLoss,
:bool => LogisticLoss,
:ord => MultinomialOrdinalLoss,
:cat => MultinomialLoss
)
and input an array of datatypes (one for each column of your data frame: GLRM(A, k, datatypes; loss_map = loss_map)
. The full call signature is
function GLRM(df::DataFrame, k::Int, datatypes::Array{Symbol,1};
loss_map = probabilistic_losses,
rx = QuadReg(.01), ry = QuadReg(.01),
offset = true, scale = false, prob_scale = true,
transform_data_to_numbers = true, NaNs_to_NAs = true)
You can modify the losses or regularizers, or turn off offsets or scaling, using these keyword arguments.
To fit a data frame with categorical values, you can use the function
expand_categoricals!
to turn categorical columns into a Boolean column for each
level of the categorical variable.
For example, expand_categoricals!(df, [:gender])
will replace the gender
column with a column corresponding to gender=male
,
a column corresponding to gender=female
, and other columns corresponding to
labels outside the gender binary, if they appear in the data set.
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!
Fitting Sparse Matrices
If you have a very large, sparsely observed dataset, then you may want to
encode your data as a
sparse matrix.
By default, LowRankModels
interprets the sparse entries of a sparse
matrix as missing entries (i.e. NA
values). There is no need to
pass the indices of observed entries (obs
)  this is done
automatically when GLRM(A::SparseMatrixCSC,...)
is called.
In addition, calling fit!(glrm)
when glrm.A
is a sparse matrix
will use the sparse variant of the proximal gradient descent algorithm,
fit!(glrm, SparseProxGradParams(); kwargs...)
.
If, instead, you'd like to interpret the sparse entries as zeros, rather
than missing or NA
entries, use:
glrm = GLRM(...;sparse_na=false)
In this case, the dataset is dense in terms of observations, but sparse
in terms of nonzero values. Thus, it may make more sense to fit the
model with the vanilla proximal gradient descent algorithm,
fit!(glrm, ProxGradParams(); kwargs...)
.
Parallel fitting (experimental)
LowRankModels makes use of Julia v0.5's new multithreading functionality to fit models in parallel. To fit a LowRankModel in parallel using multithreading, simply set the number of threads from the command line before starting Julia: eg,
export JULIA_NUM_THREADS=4
Technical details
Optimization
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.
Warm start
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 easytooptimize 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.
Initialization
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 "PCAlike" 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.init_nndsvd!
initializes the model using a modification of the NNDSVD algorithm as implemented by the NMF package. This modification handles data sets with missing entries by replacing missing entries with zeros. Optionally, by setting the argumentmax_iters=n
withn>0
, it will iteratively replace missing entries by their values as imputed by the NNDSVD, and call NNDSVD again on the new matrix. (This procedure is similar to the soft impute method of Mazumder, Hastie and Tibshirani for matrix completion.)
Parameters
As mentioned earlier, LowRankModels
uses alternating proximal
gradient descent to derive estimates of X
and Y
. This can be done
by two slightly different procedures: (A) compute the full
reconstruction, X' * Y
, to compute the gradient and objective function;
(B) only compute the model estimate for entries of A
that are observed.
The first method is likely preferred when there are few missing entries
for A
because of hardware level optimizations
(e.g. chucking the operations so they just fit in various caches). The
second method is likely preferred when there are many missing entries of
A
.
To fit with the first (dense) method:
fit!(glrm, ProxGradParams(); kwargs...)
To fit with the second (sparse) method:
fit!(glrm, SparseProxGradParams(); kwargs...)
The first method is used by default if glrm.A
is a standard
matrix/array. The second method is used by default if glrm.A
is a
SparseMatrixCSC
.
ProxGradParams()
and SparseProxGradParams()
run these respective
methods with the default parameters:
stepsize
: 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.abs_tol
: The algorithm stops when the decrease in the objective per iteration is less thanabs_tol*length(obs)
.rel_tol
: The algorithm stops when the decrease in the objective per iteration is less thanrel_tol
.max_iter
: The algorithm also stops if maximum number of roundsmax_iter
has been reached.min_stepsize
: The algorithm also stops ifstepsize
decreases below this limit.inner_iter
: specifies how many proximal gradient steps to take onX
before moving on toY
(and vice versa).
The default parameters are: ProxGradParams(stepsize=1.0;max_iter=100,inner_iter=1,abs_tol=0.00001,rel_tol=0.0001,min_stepsize=0.01*stepsize)
Convergence
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.
Imputation
After fitting a GLRM, you can use it to impute values of A
in
four different ways:
impute(glrm)
gives the maximum likelihood estimates for each entryimpute_missing(glrm)
imputes missing entries and leaves observed entries unchangedsample(glrm)
gives a draw from the posterior distribution, conditioned on the fit values ofX
andY
, for each entrysample_missing(glrm)
samples missing entries and leaves observed entries unchanged
Cross validation
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 validation

cross_validate(glrm::GLRM, nfolds=5, params=Params(); verbose=false, use_folds=None, error_fn=objective, init=None)
: performs nfold cross validation and returns average loss among all folds. More specifically, splits observations inglrm
intonfolds
groups, and builds new GLRMs, each with one group of observations left out. Fits each GLRM to the training set (the observations revealed to each GLRM) and returns the average loss on the test sets (the observations left out of each GLRM).Optional arguments:
use_folds
: builduse_folds
new GLRMs instead ofn_folds
new GLRMs, each with1/nfolds
of the entries left out. (use_folds
defaults tonfolds
.)error_fn
: use a custom error function to evaluate the fit, rather than the objective. For example, one might use the imputation error by settingerror_fn = error_metric
.init
: initialize the fit using a particular procedure. For example, considerinit=init_svd!
. See Initialization for more options.

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 (1holdout_proportion
of the original observations) and a test set (holdout_proportion
of the original observations). Performsparams.maxiter
iterations of the fitting algorithm on the training setniters
times, and returns the test and train error as a function of iteration.
Regularization paths
regularization_path(glrm::GLRM; params=Params(), reg_params=exp10.(range(2,stop=2,length=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 arrayreg_params
.
Utilities
get_train_and_test(obs, m, n, holdout_proportion=.1)
: splits observationsobs
into a train and test set.m
andn
must be at least as large as the maximal value of the first or second elements of the tuples inobservations
, respectively. Returnsobserved_features
andobserved_examples
for both train and test sets.
ScikitLearn
This library implements the
ScikitLearn.jl interface. These
models are available: SkGLRM, PCA, QPCA, NNMF, KMeans, RPCA
. See their
docstrings for more information (eg. ?QPCA
). All models support the
ScikitLearnBase.fit!
and ScikitLearnBase.transform
interface. Examples:
## Apply PCA to the iris dataset
using LowRankModels
import ScikitLearnBase
using RDatasets # may require Pkg.add("RDatasets")
A = convert(Matrix, dataset("datasets", "iris")[[:SepalLength, :SepalWidth, :PetalLength, :PetalWidth]])
ScikitLearnBase.fit_transform!(PCA(k=3, max_iter=500), A)
## Fit KMeans to a fake dataset of two Gaussians
using LowRankModels
import ScikitLearnBase
# Generate two disjoint Gaussians with 100 and 50 points
gaussian1 = randn(100, 2) + 5
gaussian2 = randn(50, 2)  10
# Merge them into a single dataset
A = vcat(gaussian1, gaussian2)
model = ScikitLearnBase.fit!(LowRankModels.KMeans(), A)
# Count how many points are assigned to each Gaussians (should be 100 and 50)
Set(sum(ScikitLearnBase.transform(model, A), 1))
See also this notebook demonstrating KMeans.
These models can be used inside a ScikitLearn pipeline, and every hyperparameter can be tuned with GridSearchCV.
Citing this package
If you use LowRankModels for published work, we encourage you to cite the software.
Use the following BibTeX citation:
@article{glrm,
title = {Generalized Low Rank Models},
author ={Madeleine Udell and Horn, Corinne and Zadeh, Reza and Boyd, Stephen},
doi = {10.1561/2200000055},
year = {2016},
archivePrefix = "arXiv",
eprint = {1410.0342},
primaryClass = "statml",
journal = {Foundations and Trends in Machine Learning},
number = {1},
volume = {9},
issn = {19358237},
url = {http://dx.doi.org/10.1561/2200000055},
}