Project: 044_On_the_Identification_of_Temporal_Causal_Represent | Function: Model.loss_function (iclr)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Implements the complete IDOL training objective combining Evidence Lower Bound (ELBO) optimization with sparsity regularization. This function computes reconstruction loss, KL divergence terms for temporal dependencies, and gradient-based sparsity penalties on Jacobian matrices to enforce sparse causal influences in the latent representation learning process.

**KEY TERMS:** {
  "sparsity_loss": "ℓsparsity sparse influence regularization",
  "kld_normal": "KL divergence for past timestamps",
  "kld_future": "KL divergence for future predictions",
  "log_pz_laplace": "prior distribution log probability"
}

**CONTEXT CODE:** 
        decomp_kernel = []
        isometric_kernel = []
        for ii in conv_kernel:
            if ii % 2 == 0: 
                decomp_kernel.append(ii + 1)
                isometric_kernel.append((configs.seq_len + configs.pred_len + ii) // ii)
            else:
                decomp_kernel.append(ii)
                isometric_kernel.append((configs.seq_len + configs.pred_len + ii - 1) // ii)

        self.task_name = configs.task_name
        self.pred_len = configs.pred_len
        self.seq_len = configs.seq_len
        
        self.decomp_multi = series_decomp_multi(decomp_kernel)

        self.z_mean = DataEmbedding(configs.enc_in, configs.d_model, configs.embed, configs.freq,
                                           configs.dropout)
        self.z_std = DataEmbedding(configs.enc_in, configs.d_model, configs.embed, configs.freq, configs.dropout)
        self.conv_trans = SeasonalPrediction(embedding_size=configs.d_model, n_heads=configs.n_heads,
                                             dropout=configs.dropout,
                                             d_layers=configs.d_layers, decomp_kernel=decomp_kernel,
                                             c_out=configs.c_out, conv_kernel=conv_kernel,
                                             isometric_kernel=isometric_kernel, device=torch.device('cpu'))

        self.regression = nn.Linear(configs.seq_len, configs.pred_len)
        self.regression.weight = nn.Parameter(
            (1 / configs.pred_len) * torch.ones([configs.pred_len, configs.seq_len]),
            requires_grad=True)

        configs.z_dim = configs.d_model
        self.rec_criterion = nn.MSELoss()
        self.decoder_dist = 'gaussian'
        self.z_dim_fix = configs.z_dim
        self.alphas_fix = SparsityMatrix(configs.z_dim)
        self.alphas_fix().requires_grad = False
        self.lag = 1
        self.transition_prior_fix = NPInstantaneousTransitionPrior(lags=self.lag,
                                                                   latent_size=configs.z_dim,
                                                                   num_layers=1,
                                                                   hidden_dim=8)
        self.register_buffer('base_dist_mean', torch.zeros(configs.z_dim))
        self.register_buffer('base_dist_var', torch.eye(configs.z_dim))

    @property
    def base_dist(self):
        # Noise density function
        return D.MultivariateNormal(self.base_dist_mean, self.base_dist_var)

    def loss_function(self, x, mus, logvars, zs):
        raise NotImplementedError("Function implementation has been masked for testing")
    
    def forward(self, x_enc, x_mark_dec, is_train=False):
        seasonal_init_enc, trend = self.decomp_multi(x_enc)
        trend = self.regression(trend.permute(0, 2, 1)).permute(0, 2, 1)

        zeros = torch.zeros([x_enc.shape[0], self.pred_len, x_enc.shape[2]], device=x_enc.device)
        seasonal_init_dec = torch.cat([seasonal_init_enc[:, -self.seq_len:, :], zeros], dim=1)
        z_mean = self.z_mean(seasonal_init_dec, x_mark_dec)
        z_std = self.z_std(seasonal_init_dec, x_mark_dec)
        
        if is_train:
            z = self.reparametrize(z_mean, z_std)
            xy = self.conv_trans(z)
            assert xy.shape[1] == (self.seq_len + self.pred_len)
            y = xy[:, -self.pred_len:, :] + trend[:, -self.pred_len:, :]

            sparsity_loss, kld_normal, kld_future = self.loss_function(xy, z_mean, z_std, z)

            x = xy[:, :self.seq_len, :]
            recon_loss = self.rec_criterion(x_enc, x)

            other_loss = self.configs.sparsity_weight * sparsity_loss + self.configs.recon_weight * recon_loss + \
                         self.configs.kld_weight * kld_normal + self.configs.kld_weight * kld_future

            return y, other_loss
        else:
            z = z_mean
            xy = self.conv_trans(z)
            assert xy.shape[1] == (self.seq_len + self.pred_len)
            y = xy[:, -self.pred_len:, :] + trend[:, -self.pred_len:, :]

            return y

    def reparametrize(self, mu, logvar):
        std = torch.exp(0.5 * logvar)
        eps = torch.randn_like(std)
        z = mu + std * eps
        return z


Complete the function implementation replacing "raise NotImplementedError".