吴恩达机器学习ex8：推荐系统

1、电影评分数据集

load ('ex8_movies.mat');

该数据集包含两个矩阵，分别是矩阵Y和矩阵R，其维度均为1682943（ij）。
Y(i,j)表示为第j个用户对第i个电影的评分，R中数据为0或1,R(i,j) = 1表示为第j个用户对第i个电影有评分，0表示为未评分。
若需要对第1部电影的已有评分计算其平均分，则代码应为：

mean(Y(1, R(1, :)));

数据可视化图形为：

2、协同过滤算法
2.1、协同过滤算法损失函数

得损失函数的计算公式为：

J = sum(sum(((X*Theta' - Y).*R).^2))/2;

运行得：

Cost at loaded parameters: 22.224604
(this value should be about 22.22)

2.2、协同过滤梯度

X_grad = R.*(X*Theta' - Y)*Theta ;
Theta_grad = (R.*(X*Theta' - Y))'*X

运行程序得：

Checking Gradients (without regularization) ... 5.5335    5.53353.6186    3.61865.4422    5.4422-1.7312   -1.73124.1196    4.1196-1.4833   -1.4833-6.0734   -6.07342.3490    2.34907.6341    7.63411.8651    1.86514.1192    4.1192-1.5834   -1.58341.2828    1.2828-6.1573   -6.15731.6628    1.66281.1686    1.16865.5630    5.56300.3050    0.30504.6442    4.6442-1.6691   -1.6691-2.1505   -2.1505-3.6832   -3.68323.4067    3.4067-4.0743   -4.07430.5567    0.5567-2.1056   -2.10560.9168    0.9168The above two columns you get should be very similar.
(Left-Your Numerical Gradient, Right-Analytical Gradient)If your cost function implementation is correct, then
the relative difference will be small (less than 1e-9). Relative Difference: 1.84952e-12

2.3、正则化损失函数

转换成代码为：

J = sum(sum((R.*(X*Theta' - Y)).^2))/2 + sum(sum(Theta.^2))*lambda/2+...sum(sum(X.^2))*lambda/2;

2.4、正则化梯度

转换成代码为：

J = sum(sum((R.*(X*Theta' - Y)).^2))/2 + sum(sum(Theta.^2))*lambda/2+...sum(sum(X.^2))*lambda/2;
X_grad = R.*(X*Theta' - Y)*Theta + X*lambda;
Theta_grad = (R.*(X*Theta' - Y))'*X + Theta*lambda;

运行程序有：

Cost at loaded parameters (lambda = 1.5): 31.344056
(this value should be about 31.34)

Checking Gradients (with regularization) ... 2.2223    2.22230.7968    0.7968-3.2924   -3.2924-0.7029   -0.7029-4.2016   -4.20163.5969    3.59690.8859    0.88591.0523    1.0523-7.8499   -7.84990.3904    0.3904-0.1347   -0.1347-2.3656   -2.36562.1066    2.10661.6703    1.67030.8519    0.8519-1.0380   -1.03802.6537    2.65370.8114    0.8114-0.8604   -0.8604-0.5884   -0.5884-0.7108   -0.7108-4.0652   -4.06520.2494    0.2494-4.3484   -4.3484-3.6167   -3.6167-4.1277   -4.1277-3.2439   -3.2439The above two columns you get should be very similar.
(Left-Your Numerical Gradient, Right-Analytical Gradient)If your cost function implementation is correct, then
the relative difference will be small (less than 1e-9). Relative Difference: 1.78901e-12

3.电影推荐系统

New user ratings:
Rated 4 for Toy Story (1995)
Rated 3 for Twelve Monkeys (1995)
Rated 5 for Usual Suspects, The (1995)
Rated 4 for Outbreak (1995)
Rated 5 for Shawshank Redemption, The (1994)
Rated 3 for While You Were Sleeping (1995)
Rated 5 for Forrest Gump (1994)
Rated 2 for Silence of the Lambs, The (1991)
Rated 4 for Alien (1979)
Rated 5 for Die Hard 2 (1990)
Rated 5 for Sphere (1998)Program paused. Press enter to continue.Training collaborative filtering...
Iteration     1 | Cost: 3.108511e+05
Iteration     2 | Cost: 1.475959e+05
Iteration     3 | Cost: 1.000321e+05
Iteration     4 | Cost: 7.707565e+04
Iteration     5 | Cost: 6.153638e+04
Iteration     6 | Cost: 5.719300e+04
Iteration     7 | Cost: 5.239113e+04
Iteration     8 | Cost: 4.771435e+04
Iteration     9 | Cost: 4.559863e+04
Iteration    10 | Cost: 4.385394e+04
Iteration    11 | Cost: 4.263562e+04
Iteration    12 | Cost: 4.184598e+04
Iteration    13 | Cost: 4.116751e+04
Iteration    14 | Cost: 4.073297e+04
Iteration    15 | Cost: 4.032577e+04
Iteration    16 | Cost: 4.009203e+04
Iteration    17 | Cost: 3.986428e+04
Iteration    18 | Cost: 3.971337e+04
Iteration    19 | Cost: 3.958890e+04
Iteration    20 | Cost: 3.949630e+04
Iteration    21 | Cost: 3.940187e+04
Iteration    22 | Cost: 3.934142e+04
Iteration    23 | Cost: 3.930822e+04
Iteration    24 | Cost: 3.926063e+04
Iteration    25 | Cost: 3.922334e+04
Iteration    26 | Cost: 3.920956e+04
Iteration    27 | Cost: 3.917145e+04
Iteration    28 | Cost: 3.914804e+04
Iteration    29 | Cost: 3.913479e+04
Iteration    30 | Cost: 3.910882e+04
Iteration    31 | Cost: 3.908992e+04
Iteration    32 | Cost: 3.908209e+04
Iteration    33 | Cost: 3.907380e+04
Iteration    34 | Cost: 3.906903e+04
Iteration    35 | Cost: 3.906437e+04
Iteration    36 | Cost: 3.905754e+04
Iteration    37 | Cost: 3.905112e+04
Iteration    38 | Cost: 3.904531e+04
Iteration    39 | Cost: 3.904023e+04
Iteration    40 | Cost: 3.903390e+04
Iteration    41 | Cost: 3.902800e+04
Iteration    42 | Cost: 3.902367e+04
Iteration    43 | Cost: 3.902195e+04
Iteration    44 | Cost: 3.902007e+04
Iteration    45 | Cost: 3.901780e+04
Iteration    46 | Cost: 3.901699e+04
Iteration    47 | Cost: 3.901489e+04
Iteration    48 | Cost: 3.901190e+04
Iteration    49 | Cost: 3.900929e+04
Iteration    50 | Cost: 3.900742e+04
Iteration    51 | Cost: 3.900630e+04
Iteration    52 | Cost: 3.900485e+04
Iteration    53 | Cost: 3.900348e+04
Iteration    54 | Cost: 3.900283e+04
Iteration    55 | Cost: 3.900208e+04
Iteration    56 | Cost: 3.900118e+04
Iteration    57 | Cost: 3.899982e+04
Iteration    58 | Cost: 3.899860e+04
Iteration    59 | Cost: 3.899710e+04
Iteration    60 | Cost: 3.899381e+04
Iteration    61 | Cost: 3.899242e+04
Iteration    62 | Cost: 3.899094e+04
Iteration    63 | Cost: 3.898986e+04
Iteration    64 | Cost: 3.898908e+04
Iteration    65 | Cost: 3.898811e+04
Iteration    66 | Cost: 3.898754e+04
Iteration    67 | Cost: 3.898736e+04
Iteration    68 | Cost: 3.898712e+04
Iteration    69 | Cost: 3.898687e+04
Iteration    70 | Cost: 3.898673e+04
Iteration    71 | Cost: 3.898634e+04
Iteration    72 | Cost: 3.898524e+04
Iteration    73 | Cost: 3.898369e+04
Iteration    74 | Cost: 3.898322e+04
Iteration    75 | Cost: 3.898257e+04
Iteration    76 | Cost: 3.898194e+04
Iteration    77 | Cost: 3.898141e+04
Iteration    78 | Cost: 3.898077e+04
Iteration    79 | Cost: 3.898025e+04
Iteration    80 | Cost: 3.897962e+04
Iteration    81 | Cost: 3.897909e+04
Iteration    82 | Cost: 3.897861e+04
Iteration    83 | Cost: 3.897735e+04
Iteration    84 | Cost: 3.897609e+04
Iteration    85 | Cost: 3.897534e+04
Iteration    86 | Cost: 3.897488e+04
Iteration    87 | Cost: 3.897468e+04
Iteration    88 | Cost: 3.897414e+04
Iteration    89 | Cost: 3.897389e+04
Iteration    90 | Cost: 3.897371e+04
Iteration    91 | Cost: 3.897355e+04
Iteration    92 | Cost: 3.897320e+04
Iteration    93 | Cost: 3.897304e+04
Iteration    94 | Cost: 3.897290e+04
Iteration    95 | Cost: 3.897276e+04
Iteration    96 | Cost: 3.897254e+04
Iteration    97 | Cost: 3.897240e+04
Iteration    98 | Cost: 3.897232e+04
Iteration    99 | Cost: 3.897222e+04
Iteration   100 | Cost: 3.897217e+04Recommender system learning completed.Program paused. Press enter to continue.Top recommendations for you:
Predicting rating 5.0 for movie Saint of Fort Washington, The (1993)
Predicting rating 5.0 for movie Great Day in Harlem, A (1994)
Predicting rating 5.0 for movie Someone Else's America (1995)
Predicting rating 5.0 for movie Entertaining Angels: The Dorothy Day Story (1996)
Predicting rating 5.0 for movie Santa with Muscles (1996)
Predicting rating 5.0 for movie Aiqing wansui (1994)
Predicting rating 5.0 for movie Prefontaine (1997)
Predicting rating 5.0 for movie They Made Me a Criminal (1939)
Predicting rating 5.0 for movie Marlene Dietrich: Shadow and Light (1996)
Predicting rating 5.0 for movie Star Kid (1997)Original ratings provided:
Rated 4 for Toy Story (1995)
Rated 3 for Twelve Monkeys (1995)
Rated 5 for Usual Suspects, The (1995)
Rated 4 for Outbreak (1995)
Rated 5 for Shawshank Redemption, The (1994)
Rated 3 for While You Were Sleeping (1995)
Rated 5 for Forrest Gump (1994)
Rated 2 for Silence of the Lambs, The (1991)
Rated 4 for Alien (1979)
Rated 5 for Die Hard 2 (1990)
Rated 5 for Sphere (1998)

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