Matrix completion and feature imputation algorithms

# fancyimpute

A variety of matrix completion and imputation algorithms implemented in Python 3.6.

## Usage

```from fancyimpute import KNN, NuclearNormMinimization, SoftImpute, BiScaler

# X is the complete data matrix
# X_incomplete has the same values as X except a subset have been replace with NaN

# Use 3 nearest rows which have a feature to fill in each row's missing features
X_filled_knn = KNN(k=3).fit_transform(X_incomplete)

# matrix completion using convex optimization to find low-rank solution
# that still matches observed values. Slow!
X_filled_nnm = NuclearNormMinimization().fit_transform(X_incomplete)

# Instead of solving the nuclear norm objective directly, instead
# induce sparsity using singular value thresholding
X_incomplete_normalized = BiScaler().fit_transform(X_incomplete)
X_filled_softimpute = SoftImpute().fit_transform(X_incomplete_normalized)

# print mean squared error for the  imputation methods above
nnm_mse = ((X_filled_nnm[missing_mask] - X[missing_mask]) ** 2).mean()
print("Nuclear norm minimization MSE: %f" % nnm_mse)

softImpute_mse = ((X_filled_softimpute[missing_mask] - X[missing_mask]) ** 2).mean()
print("SoftImpute MSE: %f" % softImpute_mse)

knn_mse = ((X_filled_knn[missing_mask] - X[missing_mask]) ** 2).mean()
print("knnImpute MSE: %f" % knn_mse)
```

## Algorithms

• `SimpleFill`: Replaces missing entries with the mean or median of each column.

• `KNN`: Nearest neighbor imputations which weights samples using the mean squared difference on features for which two rows both have observed data.

• `SoftImpute`: Matrix completion by iterative soft thresholding of SVD decompositions. Inspired by the softImpute package for R, which is based on Spectral Regularization Algorithms for Learning Large Incomplete Matrices by Mazumder et. al.

• `IterativeSVD`: Matrix completion by iterative low-rank SVD decomposition. Should be similar to SVDimpute from Missing value estimation methods for DNA microarrays by Troyanskaya et. al.

• `MatrixFactorization`: Direct factorization of the incomplete matrix into low-rank `U` and `V`, with an L1 sparsity penalty on the elements of `U` and an L2 penalty on the elements of `V`. Solved by gradient descent.

• `NuclearNormMinimization`: Simple implementation of Exact Matrix Completion via Convex Optimization by Emmanuel Candes and Benjamin Recht using cvxpy. Too slow for large matrices.

• `BiScaler`: Iterative estimation of row/column means and standard deviations to get doubly normalized matrix. Not guaranteed to converge but works well in practice. Taken from Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares.

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