1.19.0 User Documentation for Apache MADlib
Principal Component Projection
Contents

Principal component projection is a mathematical procedure that projects high dimensional data onto a lower dimensional space. This lower dimensional space is defined by the $$k$$ principal components with the highest variance in the training data.

More details on the mathematics of PCA can be found in Principal Component Analysis and some details about principal component projection calculations can be found in the Technical Background.

Projection Function
The projection functions are slightly different for dense and sparse matrices. For dense matrices:
madlib.pca_project( source_table,
pc_table,
out_table,
row_id,
residual_table,
result_summary_table
)

For sparse matrices:
madlib.pca_sparse_project( source_table,
pc_table,
out_table,
row_id,
col_id,              -- Sparse matrices only
val_id,              -- Sparse matrices only
row_dim,             -- Sparse matrices only
col_dim,             -- Sparse matrices only
residual_table,
result_summary_table
)


Arguments

source_table

TEXT. Source table name. Identical to pca_train, the input data matrix should have $$N$$ rows and $$M$$ columns, where $$N$$ is the number of data points, and $$M$$ is the number of features for each data point.

The input table for pca_project is expected to be in the one of the two standard MADlib dense matrix formats, and the sparse input table for pca_sparse_project should be in the standard MADlib sparse matrix format. These formats are described in the documentation for Principal Component Analysis.

pc_table

TEXT. Table name for the table containing principal components.

out_table

TEXT. Name of the table that will contain the low-dimensional representation of the input data.

The out_table encodes a dense matrix with the projection onto the principal components. The table has the following columns:

row_id Row id of the output matrix. A vector containing elements in the row of the matrix.

row_id

TEXT. Column name containing the row IDs in the input source table. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and N. For dense matrix format, it should contain all continguous integers from 1 to N describing the full matrix.

col_id

TEXT. Column name containing the column IDs in sparse matrix representation. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and M. This parameter applies to sparse matrices only.

val_id

TEXT. Name of 'val_id' column in sparse matrix representation defining the values of the nonzero entries. This parameter applies to sparse matrices only.

row_dim

INTEGER. The actual number of rows in the matrix. That is, if the matrix was transformed into dense format, this is the number of rows it would have. This parameter applies to sparse matrices only.

col_dim

INTEGER. The actual number of columns in the matrix. That is, if the matrix was transformed into dense format, this is the number of columns it would have. This parameter applies to sparse matrices only.

Note
The parameters 'row_dim' and 'col_dim' could actually be inferred from the sparse matrix representation, so they will be removed in the future. For now they are maintained for backward compatability so you must enter them. Making 'row_dim' or 'col_dim' larger than the actual matrix has the effect of padding it with zeros, which is probably not useful.
residual_table (optional)

TEXT, default: NULL. Name of the optional residual table.

The residual_table encodes a dense residual matrix. The table has the following columns:

row_id Row id of the output matrix. A vector containing elements in the row of the residual matrix.

result_summary_table (optional)

TEXT, default: NULL. Name of the optional summary table.

The result_summary_table contains information about the performance time of the PCA projection. The table has the following columns:

exec_time Elapsed time (ms) for execution of the function. Absolute error of the residuals. Relative error of the residuals.

Examples
SELECT madlib.pca_project();

2. Create sample data in dense matrix form:
DROP TABLE IF EXISTS mat;
CREATE TABLE mat (id integer,
row_vec double precision[]
);
INSERT INTO mat VALUES
(1, '{1,2,3}'),
(2, '{2,1,2}'),
(3, '{3,2,1}');

3. Run the PCA function for a specified number of principal components and view the results:
DROP TABLE IF EXISTS result_table, result_table_mean;
'result_table',    -- Output table
'id',              -- Row id of source table
2);               -- Number of principal components
SELECT * FROM result_table ORDER BY row_id;

 row_id |                     principal_components                     |      std_dev      |    proportion
--------+--------------------------------------------------------------+-------------------+-------------------
1 | {0.707106781186547,-6.93889390390723e-18,-0.707106781186548} |  1.41421356237309 | 0.857142857142244
2 | {0,1,0}                                                      | 0.577350269189626 | 0.142857142857041
(2 rows)

4. Project the original data to a lower dimensional representation and view the result of the projection:
DROP TABLE IF EXISTS residual_table, result_summary_table, out_table;
'result_table',
'out_table',
'id',
'residual_table',
'result_summary_table'
);
SELECT * FROM out_table ORDER BY row_id;

 row_id |               row_vec
--------+--------------------------------------
1 | {-1.41421356237309,-0.33333333333}
2 | {2.77555756157677e-17,0.66666666667}
3 | {1.41421356237309,-0.33333333333}
(3 rows)

Check the error in the projection:
SELECT * FROM result_summary_table;

   exec_time   |   residual_norm   | relative_residual_norm
---------------+-------------------+------------------------
331.792116165 | 5.89383520611e-16 |      9.68940539229e-17
(1 row)

Check the residuals:
SELECT * FROM residual_table ORDER BY row_id;

 row_id |                              row_vec
--------+--------------------------------------------------------------------
1 | {-2.22044604925031e-16,-1.11022302462516e-16,3.33066907387547e-16}
2 | {-1.12243865646685e-18,0,4.7381731349413e-17}
3 | {2.22044604925031e-16,1.11022302462516e-16,-3.33066907387547e-16}
(3 rows)

5. Now we use grouping in dense form to learn different models for different groups. First, we create sample data in dense matrix form with a grouping column. Note we actually have different matrix sizes for the different groups, which is allowed for dense:
DROP TABLE IF EXISTS mat_group;
CREATE TABLE mat_group (
id integer,
row_vec double precision[],
matrix_id integer
);
INSERT INTO mat_group VALUES
(1, '{1,2,3}', 1),
(2, '{2,1,2}', 1),
(3, '{3,2,1}', 1),
(4, '{1,2,3,4,5}', 2),
(5, '{2,5,2,4,1}', 2),
(6, '{5,4,3,2,1}', 2);

6. Run the PCA function with grouping for a specified proportion of variance and view the results:
DROP TABLE IF EXISTS result_table_group, result_table_group_mean;
'result_table_group',    -- Output table
'id',                    -- Row id of source table
0.8,                    -- Proportion of variance
'matrix_id');            -- Grouping column
SELECT * FROM result_table_group ORDER BY matrix_id, row_id;

 row_id |                                      principal_components                                      |     std_dev     |    proportion     | matrix_id
--------+------------------------------------------------------------------------------------------------+-----------------+-------------------+-----------
1 | {0.707106781186548,0,-0.707106781186547}                                                       | 1.4142135623731 | 0.857142857142245 |         1
1 | {-0.555378486712784,-0.388303582074091,0.0442457354870796,0.255566375612852,0.688115693174023} | 3.2315220311722 | 0.764102534485173 |         2
2 | {0.587384101786277,-0.485138064894743,0.311532046315153,-0.449458074050715,0.347212037159181}  |  1.795531127192 | 0.235897465516047 |         2
(3 rows)

7. Run the PCA projection on subsets of an input table based on grouping columns. Note that the parameter 'pc_table' used for projection must be generated in training using the same grouping columns.
DROP TABLE IF EXISTS mat_group_projected;
'result_table_group',
'mat_group_projected',
'id');
SELECT * FROM mat_group_projected ORDER BY matrix_id, row_id;

 row_id |                row_vec                | matrix_id
--------+---------------------------------------+-----------
1 | {1.4142135623731}                     |         1
2 | {7.40148683087139e-17}                |         1
3 | {-1.4142135623731}                    |         1
4 | {-3.59290479201926,0.559694003674779} |         2
5 | {0.924092949098971,-2.00871628417505} |         2
6 | {2.66881184290186,1.44902228049511}   |         2
(6 rows)

8. Now let's look at sparse matrices. Create sample data in sparse matrix form:
DROP TABLE IF EXISTS mat_sparse;
CREATE TABLE mat_sparse (
row_id integer,
col_id integer,
value double precision
);
INSERT INTO mat_sparse VALUES
(1, 1, 1.0),
(2, 2, 2.0),
(3, 3, 3.0),
(4, 4, 4.0),
(1, 5, 5.0),
(2, 4, 6.0),
(3, 2, 7.0),
(4, 3, 8.0);

As an aside, this is what the sparse matrix above looks like when put in dense form:
DROP TABLE IF EXISTS mat_dense;
'row=row_id, col=col_id, val=value',
'mat_dense');
SELECT * FROM mat_dense ORDER BY row_id;

 row_id |    value
--------+-------------
1 | {1,0,0,0,5}
2 | {0,2,0,6,0}
3 | {0,7,3,0,0}
4 | {0,0,8,4,0}
(4 rows)

9. Run the PCA sparse function for a specified number of principal components and view the results:
DROP TABLE IF EXISTS result_table, result_table_mean;
SELECT madlib.pca_sparse_train( 'mat_sparse',       -- Source table
'result_table',     -- Output table
'row_id',           -- Row id of source table
'col_id',           -- Column id of source table
'value',            -- Value of matrix at row_id, col_id
4,                  -- Actual number of rows in the matrix
5,                  -- Actual number of columns in the matrix
3);                 -- Number of principal components
SELECT * FROM result_table ORDER BY row_id;

Result (with principal components truncated for readability):
 row_id |         principal_components                 |     std_dev      |    proportion
--------+----------------------------------------------+------------------+-------------------
1 | {-0.0876046030186158,-0.0968983772909994,... | 4.21362803829554 | 0.436590030617467
2 | {-0.0647272661608605,0.877639526308692,...   | 3.68408023747461 | 0.333748701544697
3 | {-0.0780380267884855,0.177956517174911,...   | 3.05606908060098 | 0.229661267837836
(3 rows)

10. Project the original sparse data to low-dimensional representation:
DROP TABLE IF EXISTS mat_sparse_out;
'mat_sparse',
'result_table',
'mat_sparse_out',
'row_id',
'col_id',
'value',
4,
5
);
SELECT * FROM mat_sparse_out ORDER BY row_id;

 row_id |                         row_vec
--------+---------------------------------------------------------
1 | {4.66617015032369,-2.63552220635847,2.1865220849604}
2 | {0.228360685652383,-1.21616275892926,-4.46864627611561}
3 | {0.672067460100428,5.45249627172823,0.56445525585642}
4 | {-5.5665982960765,-1.6008113064405,1.71766893529879}
(4 rows)

11. Now we use grouping in sparse form to learn different models for different groups. First, we create sample data in sparse matrix form with a grouping column:
DROP TABLE IF EXISTS mat_sparse_group;
CREATE TABLE mat_sparse_group (
row_id integer,
col_id integer,
value double precision,
matrix_id integer);
INSERT INTO mat_sparse_group VALUES
(1, 1, 1.0, 1),
(2, 2, 2.0, 1),
(3, 3, 3.0, 1),
(4, 4, 4.0, 1),
(1, 5, 5.0, 1),
(2, 4, 6.0, 2),
(3, 2, 7.0, 2),
(4, 3, 8.0, 2);

12. Run the PCA function with grouping for a specified proportion of variance and view the results:
DROP TABLE IF EXISTS result_table_group, result_table_group_mean;
SELECT madlib.pca_sparse_train( 'mat_sparse_group',       -- Source table
'result_table_group',     -- Output table
'row_id',           -- Row id of source table
'col_id',           -- Column id of source table
'value',            -- Value of matrix at row_id, col_id
4,                 -- Actual number of rows in the matrix
5,                 -- Actual number of columns in the matrix
0.8,                 -- Proportion of variance
'matrix_id');
SELECT * FROM result_table_group ORDER BY matrix_id, row_id;

Result (with principal components truncated for readability):
 row_id |           principal_components             |     std_dev      |    proportion     | matrix_id
--------+--------------------------------------------+------------------+-------------------+-----------
1 | {-0.17805696611353,0.0681313257646983,...  | 2.73659933165925 | 0.544652792875481 |         1
2 | {-0.0492086814863993,0.149371585357526,... | 2.06058314533194 | 0.308800210823714 |         1
1 | {0,-0.479486114660443,...                  | 4.40325305087975 | 0.520500333693473 |         2
2 | {0,0.689230898585949,...                   |  3.7435566458567 | 0.376220573442628 |         2
(4 rows)

13. Projection in sparse format with grouping:
DROP TABLE IF EXISTS mat_sparse_group_projected;
'mat_sparse_group',
'result_table_group',
'mat_sparse_group_projected',
'row_id',
'col_id',
'value',
4,
5
);
SELECT * FROM mat_sparse_group_projected ORDER BY matrix_id, row_id;

 row_id |                 row_vec                 | matrix_id
--------+-----------------------------------------+-----------
1 | {-4.00039298524261,-0.626820612715982}  |         1
2 | {0.765350785238575,0.951348276645455}   |         1
3 | {1.04951017256904,2.22388180170356}     |         1
4 | {2.185532027435,-2.54840946563303}      |         1
1 | {-0.627846810195469,-0.685031603549092} |         2
2 | {-1.64754249747757,-4.7662114622896}    |         2
3 | {-3.98424961281857,4.13958468655255}    |         2
4 | {6.25963892049161,1.31165837928614}     |         2
(8 rows)


Notes
• This function is intended to operate on the principal component tables generated by pca_train or pca_sparse_train. The MADlib PCA functions generate a table containing the column-means in addition to a table containing the principal components. If this table is not found by the MADlib projection function, it will trigger an error. As long the principal component tables are created with MADlib functions, then the column-means table will be automatically found by the MADlib projection functions.
• Because of the centering step in PCA projection (see "Technical Background"), sparse matrices almost always become dense during the projection process. Thus, this implementation automatically densifies sparse matrix input, and there should be no expected performance improvement in using sparse matrix input over dense matrix input.
• Table names can be optionally schema qualified (current_schemas() is searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').
• If the input table for pca_project (pca_sparse_project) contains grouping columns, the same grouping columns must be used in the training function used to generate the principal components too.

Technical Background

Given a table containing some principal components $$\boldsymbol P$$ and some input data $$\boldsymbol X$$, the low-dimensional representation $${\boldsymbol X}'$$ is computed as

\begin{align*} {\boldsymbol {\hat{X}}} & = {\boldsymbol X} - \vec{e} \hat{x}^T \\ {\boldsymbol X}' & = {\boldsymbol {\hat {X}}} {\boldsymbol P}. \end{align*}

where $$\hat{x}$$ is the column means of $$\boldsymbol X$$ and $$\vec{e}$$ is the vector of all ones. This step is equivalent to centering the data around the origin.

The residual table $$\boldsymbol R$$ is a measure of how well the low-dimensional representation approximates the true input data, and is computed as

${\boldsymbol R} = {\boldsymbol {\hat{X}}} - {\boldsymbol X}' {\boldsymbol P}^T.$

A residual matrix with entries mostly close to zero indicates a good representation.

The residual norm $$r$$ is simply

$r = \|{\boldsymbol R}\|_F$

where $$\|\cdot\|_F$$ is the Frobenius norm. The relative residual norm $$r'$$ is

$r' = \frac{ \|{\boldsymbol R}\|_F }{\|{\boldsymbol X}\|_F }$

Related Topics
File pca_project.sql_in documenting the SQL functions