public class RowMatrix extends Object implements DistributedMatrix, Logging
| Constructor and Description |
|---|
RowMatrix(RDD<Vector> rows)
Alternative constructor leaving matrix dimensions to be determined automatically.
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RowMatrix(RDD<Vector> rows,
long nRows,
int nCols) |
| Modifier and Type | Method and Description |
|---|---|
CoordinateMatrix |
columnSimilarities()
Compute all cosine similarities between columns of this matrix using the brute-force
approach of computing normalized dot products.
|
CoordinateMatrix |
columnSimilarities(double threshold)
Compute similarities between columns of this matrix using a sampling approach.
|
CoordinateMatrix |
columnSimilaritiesDIMSUM(double[] colMags,
double gamma)
Find all similar columns using the DIMSUM sampling algorithm, described in two papers
|
MultivariateStatisticalSummary |
computeColumnSummaryStatistics()
Computes column-wise summary statistics.
|
Matrix |
computeCovariance()
Computes the covariance matrix, treating each row as an observation.
|
Matrix |
computeGramianMatrix()
Computes the Gramian matrix
A^T A. |
Matrix |
computePrincipalComponents(int k)
Computes the top k principal components.
|
SingularValueDecomposition<RowMatrix,Matrix> |
computeSVD(int k,
boolean computeU,
double rCond)
Computes singular value decomposition of this matrix.
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SingularValueDecomposition<RowMatrix,Matrix> |
computeSVD(int k,
boolean computeU,
double rCond,
int maxIter,
double tol,
String mode)
The actual SVD implementation, visible for testing.
|
RowMatrix |
multiply(Matrix B)
Multiply this matrix by a local matrix on the right.
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breeze.linalg.DenseVector<Object> |
multiplyGramianMatrixBy(breeze.linalg.DenseVector<Object> v)
Multiplies the Gramian matrix
A^T A by a dense vector on the right without computing A^T A. |
long |
numCols()
Gets or computes the number of columns.
|
long |
numRows()
Gets or computes the number of rows.
|
RDD<Vector> |
rows() |
breeze.linalg.DenseMatrix<Object> |
toBreeze()
Collects data and assembles a local dense breeze matrix (for test only).
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equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, waitinitializeIfNecessary, initializeLogging, isTraceEnabled, log_, log, logDebug, logDebug, logError, logError, logInfo, logInfo, logName, logTrace, logTrace, logWarning, logWarningpublic long numCols()
numCols in interface DistributedMatrixpublic long numRows()
numRows in interface DistributedMatrixpublic breeze.linalg.DenseVector<Object> multiplyGramianMatrixBy(breeze.linalg.DenseVector<Object> v)
A^T A by a dense vector on the right without computing A^T A.
v - a dense vector whose length must match the number of columns of this matrixpublic Matrix computeGramianMatrix()
A^T A.public SingularValueDecomposition<RowMatrix,Matrix> computeSVD(int k, boolean computeU, double rCond)
At most k largest non-zero singular values and associated vectors are returned. If there are k such values, then the dimensions of the return will be: - U is a RowMatrix of size m x k that satisfies U' * U = eye(k), - s is a Vector of size k, holding the singular values in descending order, - V is a Matrix of size n x k that satisfies V' * V = eye(k).
We assume n is smaller than m. The singular values and the right singular vectors are derived from the eigenvalues and the eigenvectors of the Gramian matrix A' * A. U, the matrix storing the right singular vectors, is computed via matrix multiplication as U = A * (V * S^-1^), if requested by user. The actual method to use is determined automatically based on the cost: - If n is small (n < 100) or k is large compared with n (k > n / 2), we compute the Gramian matrix first and then compute its top eigenvalues and eigenvectors locally on the driver. This requires a single pass with O(n^2^) storage on each executor and on the driver, and O(n^2^ k) time on the driver. - Otherwise, we compute (A' * A) * v in a distributive way and send it to ARPACK's DSAUPD to compute (A' * A)'s top eigenvalues and eigenvectors on the driver node. This requires O(k) passes, O(n) storage on each executor, and O(n k) storage on the driver.
Several internal parameters are set to default values. The reciprocal condition number rCond is set to 1e-9. All singular values smaller than rCond * sigma(0) are treated as zeros, where sigma(0) is the largest singular value. The maximum number of Arnoldi update iterations for ARPACK is set to 300 or k * 3, whichever is larger. The numerical tolerance for ARPACK's eigen-decomposition is set to 1e-10.
k - number of leading singular values to keep (0 < k <= n).
It might return less than k if
there are numerically zero singular values or there are not enough Ritz values
converged before the maximum number of Arnoldi update iterations is reached (in case
that matrix A is ill-conditioned).computeU - whether to compute UrCond - the reciprocal condition number. All singular values smaller than rCond * sigma(0)
are treated as zero, where sigma(0) is the largest singular value.public SingularValueDecomposition<RowMatrix,Matrix> computeSVD(int k, boolean computeU, double rCond, int maxIter, double tol, String mode)
k - number of leading singular values to keep (0 < k <= n)computeU - whether to compute UrCond - the reciprocal condition numbermaxIter - max number of iterations (if ARPACK is used)tol - termination tolerance (if ARPACK is used)mode - computation mode (auto: determine automatically which mode to use,
local-svd: compute gram matrix and computes its full SVD locally,
local-eigs: compute gram matrix and computes its top eigenvalues locally,
dist-eigs: compute the top eigenvalues of the gram matrix distributively)public Matrix computeCovariance()
public Matrix computePrincipalComponents(int k)
k - number of top principal components.public MultivariateStatisticalSummary computeColumnSummaryStatistics()
public RowMatrix multiply(Matrix B)
B - a local matrix whose number of rows must match the number of columns of this matrixRowMatrix representing the product,
which preserves partitioningpublic CoordinateMatrix columnSimilarities()
public CoordinateMatrix columnSimilarities(double threshold)
The threshold parameter is a trade-off knob between estimate quality and computational cost.
Setting a threshold of 0 guarantees deterministic correct results, but comes at exactly the same cost as the brute-force approach. Setting the threshold to positive values incurs strictly less computational cost than the brute-force approach, however the similarities computed will be estimates.
The sampling guarantees relative-error correctness for those pairs of columns that have similarity greater than the given similarity threshold.
To describe the guarantee, we set some notation: Let A be the smallest in magnitude non-zero element of this matrix. Let B be the largest in magnitude non-zero element of this matrix. Let L be the maximum number of non-zeros per row.
For example, for {0,1} matrices: A=B=1. Another example, for the Netflix matrix: A=1, B=5
For those column pairs that are above the threshold, the computed similarity is correct to within 20% relative error with probability at least 1 - (0.981)^10/B^
The shuffle size is bounded by the *smaller* of the following two expressions:
O(n log(n) L / (threshold * A)) O(m L^2^)
The latter is the cost of the brute-force approach, so for non-zero thresholds, the cost is always cheaper than the brute-force approach.
threshold - Set to 0 for deterministic guaranteed correctness.
Similarities above this threshold are estimated
with the cost vs estimate quality trade-off described above.public CoordinateMatrix columnSimilaritiesDIMSUM(double[] colMags, double gamma)
http://arxiv.org/abs/1206.2082 http://arxiv.org/abs/1304.1467
colMags - A vector of column magnitudesgamma - The oversampling parameter. For provable results, set to 10 * log(n) / s,
where s is the smallest similarity score to be estimated,
and n is the number of columnspublic breeze.linalg.DenseMatrix<Object> toBreeze()
DistributedMatrixtoBreeze in interface DistributedMatrix