diff options
Diffstat (limited to 'tensorflow/contrib/factorization/python/ops/factorization_ops.py')
| -rw-r--r-- | tensorflow/contrib/factorization/python/ops/factorization_ops.py | 71 |
1 files changed, 36 insertions, 35 deletions
diff --git a/tensorflow/contrib/factorization/python/ops/factorization_ops.py b/tensorflow/contrib/factorization/python/ops/factorization_ops.py index 8e0ed1d80e..811fa89bc3 100644 --- a/tensorflow/contrib/factorization/python/ops/factorization_ops.py +++ b/tensorflow/contrib/factorization/python/ops/factorization_ops.py @@ -51,9 +51,9 @@ class WALSModel(object): r"""A model for Weighted Alternating Least Squares matrix factorization. It minimizes the following loss function over U, V: - \\( - \|\sqrt W \odot (A - U V^T) \|_F^2 + \lambda (\|U\|_F^2 + \|V\|_F^2) - )\\ + $$ + \|\sqrt W \odot (A - U V^T)\|_F^2 + \lambda (\|U\|_F^2 + \|V\|_F^2) + $$ where, A: input matrix, W: weight matrix. Note that the (element-wise) square root of the weights @@ -61,12 +61,12 @@ class WALSModel(object): U, V: row_factors and column_factors matrices, \\(\lambda)\\: regularization. Also we assume that W is of the following special form: - \\( W_{ij} = W_0 + R_i * C_j )\\ if \\(A_{ij} \ne 0)\\, - \\(W_{ij} = W_0)\\ otherwise. + \\( W_{ij} = W_0 + R_i * C_j \\) if \\(A_{ij} \ne 0\\), + \\(W_{ij} = W_0\\) otherwise. where, - \\(W_0)\\: unobserved_weight, - \\(R_i)\\: row_weights, - \\(C_j)\\: col_weights. + \\(W_0\\): unobserved_weight, + \\(R_i\\): row_weights, + \\(C_j\\): col_weights. Note that the current implementation supports two operation modes: The default mode is for the condition where row_factors and col_factors can individually @@ -82,14 +82,15 @@ class WALSModel(object): normalized as follows: _, _, unregularized_loss, regularization, sum_weights = update_row_factors(sp_input) - if sp_input contains the rows {A_i, i \in I}, and the input matrix A has n - total rows, then the minibatch loss = unregularized_loss + regularization is - \\( + if sp_input contains the rows \\({A_i, i \in I}\\), and the input matrix A + has n total rows, then the minibatch loss = unregularized_loss + + regularization is + $$ (\|\sqrt W_I \odot (A_I - U_I V^T)\|_F^2 + \lambda \|U_I\|_F^2) * n / |I| + \lambda \|V\|_F^2 - )\\ + $$ The sum_weights tensor contains the normalized sum of weights - sum(W_I) * n / |I|. + \\(sum(W_I) * n / |I|\\). A typical usage example (pseudocode): @@ -223,7 +224,7 @@ class WALSModel(object): factor shard. In this case, w_ij = unobserved_weight + row_weights[i] * col_weights[j]. - If this is a single non-negative real number, this value is used for - all row weights and w_ij = unobserved_weight + row_weights * + all row weights and \\(w_ij\\) = unobserved_weight + row_weights * col_weights[j]. Note that it is allowed to have row_weights as a list while col_weights a single number or vice versa. @@ -665,18 +666,18 @@ class WALSModel(object): factors. unregularized_loss: A tensor (scalar) that contains the normalized minibatch loss corresponding to sp_input, without the regularization - term. If sp_input contains the rows {A_{i, :}, i \in I}, and the input - matrix A has n total rows, then the unregularized loss is: - (\|\sqrt W_I \odot (A_I - U_I V^T)\|_F^2 * n / |I| + term. If sp_input contains the rows \\({A_{i, :}, i \in I}\\), and the + input matrix A has n total rows, then the unregularized loss is: + \\(\|\sqrt W_I \odot (A_I - U_I V^T)\|_F^2 * n / |I|\\) The total loss is unregularized_loss + regularization. regularization: A tensor (scalar) that contains the normalized regularization term for the minibatch loss corresponding to sp_input. - If sp_input contains the rows {A_{i, :}, i \in I}, and the input matrix - A has n total rows, then the regularization term is: - \lambda \|U_I\|_F^2) * n / |I| + \lambda \|V\|_F^2. + If sp_input contains the rows \\({A_{i, :}, i \in I}\\), and the input + matrix A has n total rows, then the regularization term is: + \\(\lambda \|U_I\|_F^2) * n / |I| + \lambda \|V\|_F^2\\). sum_weights: The sum of the weights W_I corresponding to sp_input, - normalized by a factor of n / |I|. The root weighted squared error is: - \sqrt(unregularized_loss / sum_weights). + normalized by a factor of \\(n / |I|\\). The root weighted squared + error is: \sqrt(unregularized_loss / sum_weights). """ return self._process_input_helper( True, sp_input=sp_input, transpose_input=transpose_input) @@ -698,18 +699,18 @@ class WALSModel(object): factors. unregularized_loss: A tensor (scalar) that contains the normalized minibatch loss corresponding to sp_input, without the regularization - term. If sp_input contains the columns {A_{:, j}, j \in J}, and the - input matrix A has m total columns, then the unregularized loss is: - (\|\sqrt W_J \odot (A_J - U V_J^T)\|_F^2 * m / |I| + term. If sp_input contains the columns \\({A_{:, j}, j \in J}\\), and + the input matrix A has m total columns, then the unregularized loss is: + \\(\|\sqrt W_J \odot (A_J - U V_J^T)\|_F^2 * m / |I|\\) The total loss is unregularized_loss + regularization. regularization: A tensor (scalar) that contains the normalized regularization term for the minibatch loss corresponding to sp_input. - If sp_input contains the columns {A_{:, j}, j \in J}, and the input - matrix A has m total columns, then the regularization term is: - \lambda \|V_J\|_F^2) * m / |J| + \lambda \|U\|_F^2. + If sp_input contains the columns \\({A_{:, j}, j \in J}\\), and the + input matrix A has m total columns, then the regularization term is: + \\(\lambda \|V_J\|_F^2) * m / |J| + \lambda \|U\|_F^2\\). sum_weights: The sum of the weights W_J corresponding to sp_input, - normalized by a factor of m / |J|. The root weighted squared error is: - \sqrt(unregularized_loss / sum_weights). + normalized by a factor of \\(m / |J|\\). The root weighted squared + error is: \sqrt(unregularized_loss / sum_weights). """ return self._process_input_helper( False, sp_input=sp_input, transpose_input=transpose_input) @@ -720,8 +721,8 @@ class WALSModel(object): projection_weights=None): """Projects the row factors. - This computes the row embedding u_i for an observed row a_i by solving - one iteration of the update equations. + This computes the row embedding \\(u_i\\) for an observed row \\(a_i\\) by + solving one iteration of the update equations. Args: sp_input: A SparseTensor representing a set of rows. Please note that the @@ -753,8 +754,8 @@ class WALSModel(object): projection_weights=None): """Projects the column factors. - This computes the column embedding v_j for an observed column a_j by solving - one iteration of the update equations. + This computes the column embedding \\(v_j\\) for an observed column + \\(a_j\\) by solving one iteration of the update equations. Args: sp_input: A SparseTensor representing a set of columns. Please note that @@ -938,7 +939,7 @@ class WALSModel(object): loss_sp_input = (sparse_ops.sparse_transpose(new_sp_input) if transpose_input else new_sp_input) # sp_approx is the low rank estimate of the input matrix, formed by - # computing the product <u_i, v_j> for (i, j) in loss_sp_input.indices. + # computing the product <\\(u_i, v_j\\)> for (i, j) in loss_sp_input.indices. sp_approx_vals = gen_factorization_ops.masked_matmul( new_left_values, right, |