Core functions to obtain L1-L2 penalized estimates for matrix linear models.
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June 2019

MatrixLMnet: Core functions for penalized estimation for matrix linear models.

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Package for L1 and L2 penalized estimation of matrix linear models (bilinear models for matrix-valued data).

MatrixLMnet depends on the MatrixLM package, which provides core functions for closed-form least squares estimates for matrix linear models.

See the paper, "Sparse matrix linear models for structured high-throughput data", and its reproducible code for details on the L1 penalized estimation.


The MatrixLMnet package can be installed by running:

using Pkg

For the most recent version, use:

using Pkg
Pkg.add(url = "", rev="main")


using MatrixLMnet

First, construct a RawData object consisting of the response variable Y and row/column covariates X and Z. All three matrices must be passed in as 2-dimensional arrays. Note that the contr function can be used to set up treatment and/or sum contrasts for categorical variables stored in a DataFrame. By default, contr generates treatment contrasts for all specified categorical variables ("treat"). Other options include "sum" for sum contrasts, "noint" for treatment contrasts with no intercept, and "sumnoint" for sum contrasts with no intercept.

using DataFrames
using Random

# Dimensions of matrices 
n = 100
m = 250
# Number of column covariates
q = 20

# Randomly generate an X matrix of row covariates with 2 categorical variables
# and 4 continuous variables
X_df = hcat(DataFrame(catvar1=rand(1:5, n), catvar2=rand(["A", "B", "C"], n)), 
# Use the contr function to get contrasts for the two categorical variables 
# (treatment contrasts for catvar1 and sum contrasts for catvar2).
# contr returns a DataFrame, so X needs to be converted into a 2d array.
X = convert(Array{Float64,2}, contr(X_df, [:catvar1, :catvar2], 
                                    ["treat", "sum"]))
# Number of row covariates
p = size(X)[2]

# Randomly generate some data for column covariates Z and response variable Y
Z = rand(m,q)
B = rand(vcat(-5:5, zeros(19)),p,q)
E = randn(n,m)
Y = X*B*transpose(Z)+E

# Construct a RawData object
dat = RawData(Response(Y), Predictors(X, Z))

Create a 1d array of lambda penalties values to fit the estimates. If the lambdas are not in descending order, they will be automatically sorted by mlmnet.

lambdas = reverse(1.8.^(1:10))

Create a 1d array of alpha parameter penalties values that determine the penalties mix between L1 and L2 to fit the estimates according to the Elastic Net penalization method. In the case of Lasso regression (L1 regularization), alpha should be 1, and 0 for Ridge regression (L2 regularization). If the alphas are not in descending order, they will be automatically sorted by mlmnet.

alphas = reverse(collect(0:0.1:1))

L1 and L2-penalized estimates for matrix linear models can be obtained by running mlmnet. In addition to the RawData object, lambdas and alphas, mlmnet requires as a keyword argument the function name for an algorithm used to fit Elastic Net penalized estimates. Current methods are: "cd" (coordinate descent), "cd_active" (active coordinate descent), "ista" (ISTA with fixed step size), "fista" (FISTA with fixed step size), "fista_bt" (FISTA with backtracking), and "admm" (ADMM).

An object of type Mlmnet will be returned, with variables for the penalized coefficient estimates (B) along with the lambda and alpha penalty values used (lambdas, alphas). By default, mlmnet estimates both row and column main effects (X and Z intercepts), but this behavior can be suppressed by setting hasXIntercept=false and/or hasZIntercept=false; the intercepts will be regularized unless toXInterceptReg=false and/or toZInterceptReg=false. Individual X (row) and Z (column) effects can be left unregularized by manually passing in 1d boolean arrays of length p and q to indicate which effects should be regularized (true) or not (false) into toXReg and toZReg. By default, mlmnet centers and normalizes the columns of X and Z to have mean 0 and norm 1 (toNormalize=true). Additional keyword arguments include isVerbose, which controls message printing; thresh, the threshold at which the coefficients are considered to have converged; and maxiter, the maximum number of iterations.

est = mlmnet(dat, lambdas, alphas; method = "fista_bt")

If alphas argument is omitted, a Lasso regression will be applied which is equivalent to alphas = [1].

The functions for the algorithms used to fit the Elastic Net penalized estimates have keyword arguments that can be passed into mlmnet when non-default behavior is desired. Irrelevant keyword arguments will be ignored.

"cd" (coordinate descent)

  • isRandom= true: Bool; whether to use random or cyclic updates

"cd_active" (active coordinate descent)

  • isRandom = true: Bool; whether to use random or cyclic updates

"ista" (ISTA with fixed step size)

  • stepsize = 0.01: Float64; fixed step size for updates
  • setStepsize = true: Bool; whether the fixed step size should be calculated, overriding stepsize

"fista" (FISTA with fixed step size)

  • stepsize = 0.01: Float64; fixed step size for updates
  • setStepsize = true: Bool; whether the fixed step size should be calculated, overriding stepsize

"fista_bt" (FISTA with backtracking)

  • stepsize = 0.01: Float64; initial step size for updates
  • gamma = 0.5: Float64; multiplying factor for step size backtracking/line search

"admm" (ADMM)

  • rho = 1.0: Float64; parameter that controls ADMM tuning
  • setRho = true: Float64; whether the ADMM tuning parameter should be calculated, overriding rho
  • tau_incr = 2.0: Float64; parameter that controls the factor at which the ADMM tuning parameter increases
  • tau_decr = 2.0: Float64; parameter that controls the factor at which the ADMM tuning parameter decreases
  • mu = 10.0: Float64; parameter that controls the factor at which the primal and dual residuals should be within each other

The 4d array of coefficient estimates can be accessed using coef(est). Predicted values and residuals can be obtained by calling predict and resid. By default, both of these functions use the same data used to fit the model. However, a new Predictors object can be passed into predict as the newPredictors argument and a new RawData object can be passed into resid as the newData argument. For convenience, fitted(est) will return the fitted values by calling predict with the default arguments.

preds = predict(est)
resids = resid(est)

All four of these functions take optional lambda and alpha arguments, in which case only the 2d array corresponding to that values of lambda and alpha will be returned, e.g. coef(est, lambdas[1], alphas[1]). If a lambda or alpha value that was not used in the fitting of the Mlmnet object is passed in, an error will be raised. One can also extract coefficients as a flattened 3d array by calling coef_3d(est), for convenience when writing the results to flat files.

mlmnet_perms permutes the response matrix Y in a RawData object and then calls mlmnet. By default, the function used to permute Y is shuffle_rows, which shuffles the rows for Y. Alternative functions for permuting Y, such as shuffle_cols, can be passed into the argument permFun. Non-default behavior for mlmnet can be specified by passing its keyword arguments into mlmnet_perms.

estPerms = mlmnet_perms(dat, lambdas, alphas; method = "fista_bt")

Cross-validation for mlmnet is implemented by mlmnet_cv. The user can either manually specify the row/column folds of Y as a 1d array of 1d arrays of row/column indices, or specify the number of folds that should be used. If the number of folds is specified, disjoint folds of approximately equal size will be generated from a call to make_folds. Passing in 1 for the number of row (or column) folds indicates that all of the rows (or columns) of Y should be used in each fold. The advantage of manually passing in the row and/or column folds is that it allows the user to stratify or otherwise control the nature of the folds. For example, make_folds_conds will generate folds for a set of categorical conditions and ensure that each condition is represented in each fold. Cross validation is computed in parallel when possible. Non-default behavior for mlmnet can be specified by passing its keyword arguments into mlmnet_cv.

In the call below, mlmnet_cv generates 10 disjoint row folds but uses all columns of Y in each fold (indicated by the 1). The function returns an Mlmnet_cv object, which contains an array of the Mlmnet objects for each fold (MLMNets); the lambda penalty values used (lambdas); the row and column folds (rowFolds and colFolds); an array of the mean-squared error for each fold (mse); and an array of the proportion of zero interaction effects for each fold (propZero). The keyword argument dig in mlmnet_cv adjusts the level of precision when calculating the percent of zero coefficients. It defaults to 12.

estCVObjs = mlmnet_cv(dat, lambdas, alphas, 10, 1, method = "fista_bt")

mlmnet_cv_summary displays a table of the average mean-squared error and proportion of zero coefficients across the folds for each value of lambda. The optimal lambda might be the one that minimizes the mean-squared error (MSE), or can be chosen based on a pre-determined proportion of zeros that is desired in the coefficient estimates.


The lambda_min function returns the summary information for the lambdas that correspond to the minimum average test MSE across folds and the MSE that is one standard error greater.


Additional details can be found in the documentation for specific functions.

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