低秩矩阵分解Low-rank Matrix Factorization

这个模块应用因子模型”factor model”来使用一个低秩近似矩阵low-rank approximation来表示一个不完全矩阵incomplete matrix.数学上，该模型致力于对任意给定的不完全矩阵，找到最小的矩阵U和V（也被称为因子factor），：低秩矩阵分解 - 图1 并服从低秩矩阵分解 - 图2.其中低秩矩阵分解 - 图3表示Frobenius范数，令A是一个m x n矩阵，U是 m x r，V是 n x r,in dimension，并且低秩矩阵分解 - 图4。该模型没有打算实现完全分解full decomposition，或被用于reverse步骤的一部分。该模型被广泛用于推荐系统recommendation system（例如Netflix）及特征选择feature selection（例如图像处理image process）。

函数语法：

不完全矩阵incomplete matrix使用低秩矩阵分解成两个因子factor

  1. lmf_igd_run( rel_output,
  2.          rel_source,
  3.          col_row,
  4.          col_column,
  5.          col_value,
  6.          row_dim,
  7.          column_dim,
  8.          max_rank,
  9.          stepsize,
  10.          scale_factor,
  11.          num_iterations,
  12.          tolerance
  13.        )

参数

rel_output

 TEXT，输出的表名  输出factor matrix U和V是flattened format

  1. RESULT AS (
  2.     matrix_u    DOUBLE PRECISION[],
  3.     matrix_v    DOUBLE PRECISION[],
  4.     rmse        DOUBLE PRECISION

);
row i相当于matrix_u[i:i][1:r]，column j相当于matrix_v[j:j][1:r].

rel_source

TEXT. 包含输入数据的表名.

输入 matrix> 需要是以下格式:

  1. {TABLE|VIEW} input_table (
  2. row    INTEGER,
  3. col    INTEGER,
  4. value  DOUBLE PRECISION
  5. )

输入数据（不完全矩阵incomplete matrix）包含在一张表中，矩阵中的有效entity被定义为(row, column, value).输入矩阵必须为1-based，意思是row >= 1, 和 col >= 1. NULL值不被允许

col_row

TEXT. 包含 row number的列名.

col_column

TEXT. 包含 column number的列名.

col_value

DOUBLE PRECISION. The value at (row, col).

row_dim (optional)

INTEGER, default: “SELECT max(col_row) FROM rel_source”. 矩阵的列数.

column_dim (optional)

INTEGER, default: “SELECT max(col_col) FROM rel_source”. 矩阵的行数.

max_rank

INTEGER, default: 20. 近似矩阵的秩（The rank of desired approximation）.

stepsize (optional)

DOUBLE PRECISION, default: 0.01. Hyper-parameter that decides how aggressive the gradient steps are.

scale_factor (optional)

DOUBLE PRECISION, default: 0.1.决定初始因子factor范围scale的Hyper-parameter （Hyper-parameter that decides scale of initial factors）.

num_iterations (optional)

INTEGER, default: 10. 最大迭代次数无论是否收敛（Maximum number if iterations to perform regardless of convergence）.

tolerance (optional)

DOUBLE PRECISION, default: 0.0001.收敛时可接受的误差等级（Acceptable level of error in convergence）.

例子：
准备输入表： 

  1. DROP TABLE IF EXISTS lmf_data;
  2. CREATE TABLE lmf_data (
  3.  row INT,
  4.  col INT,
  5. val FLOAT8
  6. );

插入数据: 

  1. INSERT INTO lmf_data VALUES (1, 1, 5.0);  
  2. INSERT INTO lmf_data VALUES (3, 100, 1.0);  
  3. INSERT INTO lmf_data VALUES (999, 10000, 2.0);

调用lmf_igd_run()存储过程 

  1. DROP TABLE IF EXISTS lmf_model;  
  2. SELECT madlib.lmf_igd_run(
  3.                        'lmf_model',
  4.                        'lmf_data',
  5.                        'row',
  6.                        'col',
  7.                        'val',
  8.                        999,
  9.                        10000,
  10.                        3,
  11.                        0.1,
  12.                        2,
  13.                        10,
  14.                        1e-9
  15.                      );

结果为：

NOTICE: Matrix lmf_data to be factorized: 999 x 10000
NOTICE:
Finished low-rank matrix factorization using incremental gradient
table : lmf_data (row, col, val)
Results:
RMSE = 2.55605187132731e-06
Output:
view : SELECT * FROM lmf_model WHERE id = 1
lmf_igd_run
1
(1 row)

要查看结果，你需要返回的model id(上面lmf_igd_run()函数已经指出id为1)

SELECT array_dims(matrix_u) AS u_dims,array_dims(matrix_v) AS v_dims
FROM lmf_model
WHERE id = 1;
u_dims | v_dims
———————+————————
[1:999][1:3] | [1:10000][1:3]
(1 row)

查询结果值

SELECT matrix_u[2:2][1:3] AS row_2_features
FROM lmf_model
WHERE id = 1; 

  1.                   row_2_features

{{0.296678517945111,1.14573554042727,0.317661632783711}}
(1 row)

预估缺失entity

SELECT matrix_u[2:2][1:3] AS row_2_features
FROM lmf_model
WHERE id = 1; 

  1.                   row_2_features

{{0.296678517945111,1.14573554042727,0.317661632783711}}
(1 row)

参考资料

[1] N. Srebro and T. Jaakkola. “Weighted Low-Rank Approximations.” In: ICML. Ed. by T. Fawcett and N. Mishra. AAAI Press, 2003, pp. 720–727. isbn: 1-57735-189-4.

[2] Simon Funk, Netflix Update: Try This at Home, December 11 2006, http://sifter.org/~simon/journal/20061211.html

[3] J. Wright, A. Ganesh, S. Rao, Y. Peng, and Y. Ma. “Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization.” In: NIPS. Ed. by Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta. Curran Associates, Inc., 2009, pp. 2080–2088. isbn: 9781615679119.