A Julia package for non-negative matrix factorization (NMF).

**Note:** Nonnegative Matrix Factorization is an area of active research. New algorithms are proposed every year. Contributions are very welcomed.

- Lee & Seung's Multiplicative Update (for both MSE & Divergence objectives)
- (Naive) Projected Alternate Least Squared
- ALS Projected Gradient Methods
- Random Initialization
- NNDSVD Initialization

- Sparse NMF
- Probabilistic NMF

*Non-negative Matrix Factorization (NMF)* generally refers to the techniques for factorizing a non-negative matrix `X`

into the product of two lower rank matrices `W`

and `H`

, such that `WH`

optimally approximates `X`

in some sense. Such techniques are widely used in text mining, image analysis, and recommendation systems.

This package provides two sets of tools, respectively for *initilization* and *optimization*. A typical NMF procedure consists of two steps: (1) use an initilization function that initialize `W`

and `H`

; and (2) use an optimization algorithm to pursue the optimal solution.

Most types and functions (except the high-level function `nnmf`

) in this package are not exported. Users are encouraged to use them with the prefix `NMF.`

. This way allows us to use shorter names within the package and makes the codes more explicit and clear on the user side.

The package provides a high-level function `nnmf`

that runs the entire procedure (initialization + optimization):

**nnmf**(X, k, ...)

This function factorizes the input matrix `X`

into the product of two non-negative matrices `W`

and `H`

.

In general, it returns a result instance of type `NMF.Result`

, which is defined as

```
struct Result
W::Matrix{Float64} # W matrix
H::Matrix{Float64} # H matrix
niters::Int # number of elapsed iterations
converged::Bool # whether the optimization procedure converges
objvalue::Float64 # objective value of the last step
end
```

The function supports the following keyword arguments:

`init`

: A symbol that indicates the initialization method (default =`:nndsvdar`

).This argument accepts the following values:

`random`

: matrices filled with uniformly random values`nndsvd`

: standard version of NNDSVD`nndsvda`

: NNDSVDa variant`nndsvdar`

: NNDSVDar variant

`alg`

: A symbol that indicates the factorization algorithm (default =`:alspgrad`

).This argument accepts the following values:

`multmse`

: Multiplicative update (using MSE as objective)`multdiv`

: Multiplicative update (using divergence as objective)`projals`

: (Naive) Projected Alternate Least Square`alspgrad`

: Alternate Least Square using Projected Gradient Descent

`maxiter`

: Maximum number of iterations (default =`100`

).`tol`

: tolerance of changes upon convergence (default =`1.0e-6`

).`verbose`

: whether to show procedural information (default =`false`

).

**NMF.randinit**(X, k[; zeroh=false, normalize=false])Initialize

`W`

and`H`

given the input matrix`X`

and the rank`k`

. This function returns a pair`(W, H)`

.Suppose the size of

`X`

is`(p, n)`

, then the size of`W`

and`H`

are respectively`(p, k)`

and`(k, n)`

.Usage:

`W, H = NMF.randinit(X, 3)`

For some algorithms (

*e.g.*ALS), only`W`

needs to be initialized. For such cases, one may set the keyword argument`zeroh`

to be`true`

, then in the output`H`

will be simply a zero matrix of size`(k, n)`

.Another keyword argument is

`normalize`

. If`normalize`

is set to`true`

, columns of`W`

will be normalized such that each column sum to one.**NMF.nndsvd**(X, k[; zeroh=false, variant=:std])Use the

*Non-Negative Double Singular Value Decomposition (NNDSVD)*algorithm to initialize`W`

and`H`

.Reference: C. Boutsidis, and E. Gallopoulos. SVD based initialization: A head start for nonnegative matrix factorization. Pattern Recognition, 2007.

Usage:

`W, H = NMF.nndsvd(X, k)`

This function has two keyword arguments:

`zeroh`

: have`H`

initialized when it is set to`true`

, or set`H`

to all zeros when it is set to`false`

.`variant`

: the variant of the algorithm. Default is`std`

, meaning to use the standard version, which would generate a rather sparse`W`

. Other values are`a`

and`ar`

, respectively corresponding to the variants:*NNDSVDa*and*NNDSVDar*. Particularly,`ar`

is recommended for dense NMF.

This package provides multiple factorization algorithms. Each algorithm corresponds to a type. One can create an algorithm *instance* by choosing a type and specifying the options, and run the algorithm using `NMF.solve!`

:

**NMF.solve!**(alg, X, W, H)

Use the algorithm `alg`

to factorize `X`

into `W`

and `H`

.

Here, `W`

and `H`

must be pre-allocated matrices (respectively of size `(p, k)`

and `(k, n)`

). `W`

and `H`

must be appropriately initialized before this function is invoked. For some algorithms, both `W`

and `H`

must be initialized (*e.g.* multiplicative updating); while for others, only `W`

needs to be initialized (*e.g.* ALS).

The matrices `W`

and `H`

are updated in place.

**Multiplicative Updating**Reference: Daniel D. Lee and H. Sebastian Seung. Algorithms for Non-negative Matrix Factorization. Advances in NIPS, 2001.

This algorithm has two different kind of objectives: minimizing mean-squared-error (

`:mse`

) and minimizing divergence (`:div`

). Both`W`

and`H`

need to be initialized.`MultUpdate(obj=:mse, # objective, either :mse or :div maxiter=100, # maximum number of iterations verbose=false, # whether to show procedural information tol=1.0e-6, # tolerance of changes on W and H upon convergence lambda=1.0e-9) # regularization coefficients (added to the denominator)`

**Note:**the values above are default values for the keyword arguments. One can override part (or all) of them.**(Naive) Projected Alternate Least Square**This algorithm alternately updates

`W`

and`H`

while holding the other fixed. Each update step solves`W`

or`H`

without enforcing the non-negativity constrait, and forces all negative entries to zeros afterwards. Only`W`

needs to be initialized.`ProjectedALS(maxiter::Integer=100, # maximum number of iterations verbose::Bool=false, # whether to show procedural information tol::Real=1.0e-6, # tolerance of changes on W and H upon convergence lambda_w::Real=1.0e-6, # L2 regularization coefficient for W lambda_h::Real=1.0e-6) # L2 regularization coefficient for H`

**Alternate Least Square Using Projected Gradient Descent**Reference: Chih-Jen Lin. Projected Gradient Methods for Non-negative Matrix Factorization. Neural Computing, 19 (2007).

This algorithm adopts the alternate least square strategy. A efficient projected gradient descent method is used to solve each sub-problem. Both

`W`

and`H`

need to be initialized.`ALSPGrad(maxiter::Integer=100, # maximum number of iterations (in main procedure) maxsubiter::Integer=200, # maximum number of iterations in solving each sub-problem tol::Real=1.0e-6, # tolerance of changes on W and H upon convergence tolg::Real=1.0e-4, # tolerable gradient norm in sub-problem (first-order optimality) verbose::Bool=false) # whether to show procedural information`

Here are examples that demonstrate how to use this package to factorize a non-negative dense matrix.

```
... # prepare input matrix X
r = nnmf(X, k; alg=:multmse, maxiter=30, tol=1.0e-4)
W = r.W
H = r.H
```

```
import NMF
# initialize
W, H = NMF.randinit(X, 5)
# optimize
NMF.solve!(NMF.MultUpdate(obj=:mse,maxiter=100), X, W, H)
```

```
import NMF
# initialize
W, H = NMF.randinit(X, 5)
# optimize
NMF.solve!(NMF.ProjectedALS(maxiter=50), X, W, H)
```

```
import NMF
# initialize
W, H = NMF.nndsvdar(X, 5)
# optimize
NMF.solve!(NMF.ALSPGrad(maxiter=50, tolg=1.0e-6), X, W, H)
```

02/08/2014

16 days ago

54 commits