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- How to store the matrices?
- How to perform matrix operations using MapReduce?
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Introduction
Hama (Hadoop Matrix) is a distributed matrix computation package currently in incubation with Apache. It is a library of matrix operations for large-scale processing and development environments as well as a Map/Reduce framework for a large-scale numerical analysis and data mining, that need the intensive computation power of matrix inversion, e.g., linear regression, PCA, SVM and etc. It will be useful for many scientific applications, e.g., physics computations, linear algebra, computational fluid dynamics, statistics, graphic rendering and many more.
Block Diagram
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\[http://wiki.apache.org/hama-data/attachments/Architecture/attachments/block.png\] |
Implementation
User Interfaces
Storage Structure
Store Matrices & Graphs
The matrix or network structure that frequently changes should have flexible storage structure for easy update and indicies that point to the appropriate entry. Also, we need a model that uses the concept of column-iterative methods.
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– Just a thought, considering the depleted activity in HBase should we not explore ways to avoid HBase ? --Prasen
– Hbase seems activity at this time, However Yes. We should think about it. --Edward
Structure Considerations
A lot of columns causes huge storage expense. So I propose that we store a piece of Vector to each cell.
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SparseMatrix Table scheme:
column: metadata:
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row1 column:column1 <entry1> metadata:matrixType <SparseMatrix>
column:column2 <entry2> ...
column:column3 <entry3>
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row2
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Perform matrix operations
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Algorithms
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The Hadoop/Hbase is designed to efficiently process large data set by connecting many commodity computers together to work in parallel but, If there's a inter-node communication, the elapsed run time will be slower with more nodes. Consequently, an "effective" algorithm should avoid large amounts of communication.
Dense Matrix-Matrix addition
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Basic Algorithms
Addition
Addition of multiple matrices
Matrix-Matrix multiplication
Dense Matrices multiplication
Multiplication
- Iterative Approach
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For i = 0 step 1 until N -1 Job: Computes the ith row of C = Matrix-Vector multiplication Iterative job: - A map task receives a row n of B as a key, and vector of row as its value - Multiplying by all columns of ith row of A - Reduce task find and add the ith product 1st + + + + | a11 a12 a13 | | a11 a21 a31 | | ... ... ... | X | a12 a22 a32 | | ... ... ... | | a13 a23 a33 | + + + + 2nd + + + + | ... ... ... | | a11 a21 a31 | | a21 a22 a23 | X | a12 a22 a32 | | ... ... ... | | a13 a23 a33 | + + + + .... |
- Blocking Algorithm Approach
To mutliply two dense matrices A and B, We collect the blocks to 'collectionTable' firstly using map/reduce. Rows are named as c(i, j) with sequential number ((N^2 * i) + ((j * N) + k) to avoid duplicated records.
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Blocking jobs: Collect the blocks to 'collectionTable' from A and B. - A map task receives a row n as a key, and vector of each row as its value - emit (blockID, sub-vector) pairs - Reduce task merges block structures based on the information of blockID Multiplication job: - A map task receives a blockID n as a key, and two sub-matrices of A and B as its value - Multiply two sub-matrices: a[i][j] * b[j][k] - Reduce task computes sum of blocks - c[i][k] += multiplied blocks |
– If A stored on Hbase and we collect the B to A, Can we reduce the run time of blocking job. --Edward
– Hmm, It's possible to blocking at once. --Edward
Sparse Matrices multiplication
Matrix Norm
- Find the maximum absolute row sum of matrix
Matrix.Norm.One is that find the maximum absolute row sum of matrix. Comparatively, it's a good fit with MapReduce model because doesn't need iterative jobs or table/file JOIN operations.
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j=n
The maximum absolute row sum = max ( sum | a_{i,j} | )
1<=i<=n j=1
- A map task receives a row n as a key, and vector of each row as its value
- emit (row, the sum of the absolute value of each entries)
- Reduce task select the maximum one
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Compute the infinity norm
It's a maximum absolute column sum of matrix.
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i=n
The maximum absolute column sum = max ( sum | a_{i,j} | )
1<=j<=n i=1
- A map task receives a row n as a key, and vector of each row as its value
- emit (column, the absolute value of each entries)
- Reduce task sum the absolute values of column, select the one maximum one.
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NOTE: NOTE: Matrix.infinity, Matrix.Maxvalue and Matrix.Frobenius are almost same with this.
Compute the transpose of matrix
The transpose of a matrix is another matrix in which the rows and columns have been reversed. The matrix must be square for this work.
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+ + + + | a11 a12 a13 | | a11 a21 a31 | | a21 a22 a23 | => | a12 a22 a32 | | a31 a32 a33 | | a13 a23 a33 | + + + + - A map task receives a row n as a key, and vector of each row as its value - emit (Reversed index, the entry with the given index) - Reduce task sets the reversed values |
Compute the determinant of square matrix
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Decomposition Algorithms
Cholesky Decomposition
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