numpy lstm

### 使用NumPy实现LSTM神经网络

为了理解如何使用NumPy来构建一个简单的LSTM单元，可以考虑以下简化版本的LSTM模型。此方法有助于深入理解LSTM的工作原理及其内部机制。

#### 初始化权重矩阵和偏置向量
定义函数用于初始化LSTM所需的权重矩阵和偏置项：

def initialize_parameters(n_x, n_a):
    """
    Initialize parameters with small random values.
    
    Arguments:
    n_x -- number of units in the input vector (dimensionality of a single time step data point)
    n_a -- number of LSTM units
    
    Returns:
    params -- python dictionary containing initialized weights and biases
    """

    Wf = np.random.randn(n_a, n_a + n_x) * 0.01  # Forget gate weight matrix
    bf = np.zeros((n_a, 1))                      # Forget gate bias vector
    Wi = np.random.randn(n_a, n_a + n_x) * 0.01  # Input update gate weight matrix
    bi = np.zeros((n_a, 1))                      # Input update gate bias vector
    Wo = np.random.randn(n_a, n_a + n_x) * 0.01  # Output gate weight matrix
    bo = np.zeros((n_a, 1))                      # Output gate bias vector
    Wc = np.random.randn(n_a, n_a + n_x) * 0.01  # Candidate cell state weight matrix
    bc = np.zeros((n_a, 1))                      # Candidate cell state bias vector

    params = {"Wf": Wf, "bf": bf,
              "Wi": Wi, "bi": bi,
              "Wo": Wo, "bo": bo,
              "Wc": Wc, "bc": bc}

    return params

#### 单步前向传播过程
接下来是单个时间步骤内的前向计算逻辑，它接收当前时刻输入xt及时刻t−1的状态ht−1和记忆ct−1作为输入，并输出新的状态ht和更新后的记忆ct:

def lstm_cell_forward(xt, a_prev, c_prev, parameters):
    """
    Implement the forward propagation for the LSTM-cell.

    Arguments:
    xt -- your input data at timestep "t", numpy array of shape (n_x, m).
    a_prev -- Hidden state at timestep "t-1", numpy array of shape (n_a, m)
    c_prev -- Memory state at timestep "t-1", numpy array of shape (n_a, m)
    parameters -- Weight matrix of the forget gate, numpy array of shape (n_a, n_a + n_x)
                        bf -- Bias of the forget gate, numpy array of shape (n_a, 1)
                        ...
                        
    Returns:
    a_next -- next hidden state, of shape (n_a, m)
    c_next -- next memory state, of shape (n_a, m)
    yt_pred -- prediction at timestep "t", numpy array of shape (n_y, m)
    cache -- tuple of values needed for the backward pass, contains (a_next, c_next, a_prev, c_prev, ft, it, ct_hat, ot, xt, parameters)
    """

    # Retrieve parameters from "parameters"
    Wf = parameters["Wf"]
    bf = parameters["bf"]
    Wi = parameters["Wi"]
    bi = parameters["bi"]
    Wo = parameters["Wo"]
    bo = parameters["bo"]
    Wc = parameters["Wc"]
    bc = parameters["bc"]

    # Concatenate a_prev and xt to get concatenated input
    concat = np.concatenate([a_prev, xt], axis=0)

    # Compute all gates using activation functions sigmoid or tanh where appropriate
    ft = sigmoid(np.dot(Wf, concat) + bf)        # Forget gate activations
    it = sigmoid(np.dot(Wi, concat) + bi)        # Update gate activations
    cct = np.tanh(np.dot(Wc, concat) + bc)       # Cell candidate value
    c_next = ft * c_prev + it * cct             # New cell state
    ot = sigmoid(np.dot(Wo, concat) + bo)        # Output gate activations
    a_next = ot * np.tanh(c_next)               # Next hidden state

    # Define predictions as output layer's result after applying softmax function on top of 'ot'
    yt_pred = None                              # Placeholder; actual implementation depends on task requirements

    # Store intermediate results into `cache` variable which will be used during backpropagation phase later
    cache = (a_next, c_next, a_prev, c_prev, ft, it, cct, ot, xt, parameters)

    return a_next, c_next, yt_pred, cache

上述代码片段展示了如何利用NumPy库创建并执行一次完整的LSTM细胞操作[^1]。需要注意的是，在实际应用中还需要编写反向传播算法以便能够调整参数从而最小化损失函数；此外，对于具体的预测任务还需设计相应的输出层结构。