This effort is still a "work in progress". Please feel free to add comments. BRBut please make the content less visible by using smaller fonts. – Edward J. Yoon
Overview
This is intended to explain and to illustrate the concept of Hama. There are two main parts:
- How to store the matrices?
- How to perform matrix operations using MapReduce?
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
[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.
HBase is an open-source, distributed, column-oriented store modeled as a google bigtable. Hama uses [http://hadoop.apache.org/hbase/ Hbase] to store the matrices and graphs which are represented mathematically.
Matrices are basically tables. They are ways of storing numbers and other things. Typical matrix has rows and columns which is often called a 2-way matrix because it has two dimensions. For example, you might have respondents-by-attitudes. Of course, you might collect the same data on the same people at 5 points in time. In that case, you either have 5 different 2-way matrices, or you could think of it as a 3-way matrix, that is respondent-by-attitude-by-time.
– 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.
DenseMatrix Table scheme:
column: metadata:
===============================================================================
row1 column:startLocation <sub-vector1> metadata:subVectorInterval <1000>
column:startLocation <sub-vector2> metadata:matrixType <DenseMatrix>
column:startLocation <sub-vector3> ...
...
row2
...
SparseMatrix Table scheme:
column: metadata:
===============================================================================
row1 column:column1 <entry1> metadata:matrixType <SparseMatrix>
column:column2 <entry2> ...
column:column3 <entry3>
...
row2
...
Algorithms
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.
Basic Algorithms
Addition
Addition of multiple matrices
Multiplication
- Iterative Approach
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.
CollectionTable:
matrix A matrix B
------------------------+-------------------------------
block(0, 0)-0 block(0, 0) block(0, 0)
block(0, 0)-1 block(0, 1) block(1, 0)
block(0, 0)-2 block(0, 2) block(2, 0)
... N ...
block(N-1, n-1)-(N^3-1) block(N-1, N-1) block(N-1, N-1)
Each row has a two sub matrices of a(i, k) and b(k, j) so that minimized data movement and network cost.
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
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.
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
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.
+ + + + | 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