Matrix multiplication pipeline

Reconstruct the Olden's-method feature attribution score matrix from trained MiMeNet networks

Summary

Extract weight matrices from trained neural network models and compute microbe-metabolite feature attribution scores using Olden's method (successive layer-wise weight matrix multiplication). This skill produces normalized, clipped interaction score matrices that identify which microbial features drive predictions of specific metabolomic features.

When to use

After completing 10-fold cross-validated training of MiMeNet neural networks on paired microbiome-metabolome datasets and identifying well-predicted metabolites (those with Spearman correlation coefficient above the 95th percentile of background distribution). Use this skill when you need interpretable microbe-metabolite interaction patterns to seed downstream biclustering and module discovery.

When NOT to use

• Single trained model or <10 cross-validated model instances (insufficient sampling of weight-space variability)
• When metabolite predictions have not yet been filtered by Spearman correlation threshold (attribution scores would include noise from poorly-predicted metabolites)
• When input features have been filtered to <10% sample prevalence but feature names/indices have not been synchronized across training and attribution extraction

Inputs

• Trained neural network models (100 total from 10 iterations of 10-fold cross-validation)
• Weight matrices W_l extracted from all hidden layers of each model
• Background distribution of attribution scores from shuffled-data cross-validation
• 97.5th percentile threshold value for significance determination
• List of well-predicted metabolites (SCC > 95th percentile)

Outputs

• 100 per-model microbe-metabolite attribution score matrices (one per trained network)
• Normalized and clipped attribution matrices with values in [-1, +1]
• Filtered attribution matrices containing only microbes with ≥1 significant score
• Structured output format (NumPy arrays or CSV tables indexed by microbe and metabolite feature names)

How to apply

Load all trained neural network weight matrices W_l from hidden layers l across all 100 models (10 iterations × 10 folds). For each trained model, compute the product of successive layer weights: S = ∏(W_l) where rows represent microbe input features and columns represent metabolite output features. Positive scores indicate increased microbe abundance predicts increased metabolite abundance; negative scores indicate inverse relationship. Normalize each attribution matrix S_i by dividing by the 97.5th percentile threshold identified from the background shuffled-data distribution, then clip all values to the range [-1, +1]. Retain only rows (microbes) with at least one significant attribution score (absolute value > 97.5th percentile threshold) for downstream use.

Related tools

• MiMeNet (Neural network framework that trains multilayer perceptron models on microbiome-metabolome data and provides trained weight matrices for attribution extraction) — https://github.com/YDaiLab/MiMeNet
• TensorFlow or PyTorch (Deep learning backend for loading trained model weights and performing matrix operations)
• NumPy (Matrix multiplication (np.prod or np.matmul) and element-wise normalization/clipping operations)
• Pandas (DataFrame-based I/O and feature name indexing for attributed score matrices)

Examples

# Load 100 trained MiMeNet models and compute Olden's attributions
for model_idx in range(100):
    model = load_trained_model(f'models/cv_fold_{model_idx}.h5')
    weights = [model.layers[i].get_weights()[0] for i in range(model.n_hidden_layers)]
    S = np.linalg.multi_dot(weights)  # Olden's method: successive matrix product
    S_norm = S / background_threshold_97p5
    S_clipped = np.clip(S_norm, -1, 1)
    S_filtered = S_clipped[microbe_indices_with_significant_scores, :]
    attribution_matrices.append(S_filtered)

Evaluation signals

• All 100 attribution matrices have shape (n_microbes, n_metabolites) matching training data dimensions
• After normalization and clipping, 100% of values fall within [-1.0, +1.0] range
• Distribution of attribution score absolute values in each matrix exhibits expected bimodal pattern: background noise near zero and signal tail above 97.5th percentile threshold
• Number of retained microbes (rows with ≥1 significant score) matches the count reported in cross-validation results (e.g., '163 microbes with at least one significant attribution score')
• Symmetry check: sign of attribution score is consistent with direction of microbe–metabolite correlation in the original trained model predictions

Limitations

• Attribution scores reflect learned network weights but do not incorporate mechanistic or biochemical knowledge; high scores do not prove causal relationships
• Olden's method assumes multiplicative composition of layer-wise effects; non-linear layer interactions (e.g., from ReLU activation functions) are not explicitly captured in the product
• Attribution matrices are data-driven and may reflect confounding or indirect associations rather than direct microbe-metabolite interactions
• Performance depends on quality of upstream training; if metabolite predictions are poor (low Spearman correlation), resulting attribution scores are unreliable noise
• Threshold choice (97.5th percentile for significance) is empirically determined from background distribution but may require adjustment for datasets with different sample sizes or feature richness

Evidence

• [other] Load trained neural network models from 10 iterations of 10-fold cross-validation (100 total models). For each trained model, extract weight matrices W_l from all hidden layers l in the network.: "Load trained neural network models from 10 iterations of 10-fold cross-validation (100 total models). For each trained model, extract weight matrices W_l from all hidden layers l in the network."
• [other] Apply Olden's method by computing the product of weight matrices across all layers: S = ∏(W_l) for l ∈ L, where each row represents a microbe input feature and each column represents a metabolite output feature.: "Apply Olden's method by computing the product of weight matrices across all layers: S = ∏(W_l) for l ∈ L, where each row represents a microbe input feature and each column represents a metabolite"
• [other] positive scores indicate increased microbe abundance leads to increased metabolite abundance, and negative scores indicate increased microbe abundance leads to decreased metabolite abundance: "positive scores indicate increased microbe abundance leads to increased metabolite abundance, and negative scores indicate increased microbe abundance leads to decreased metabolite abundance"
• [results] MiMeNet constructs a score matrix of microbe-metabolite feature attributions between the microbes and well-predicted metabolites using the learned network weights obtained from cross-validation training: "constructs a score matrix of microbe-metabolite feature attributions between the microbes and well-predicted metabolites using the learned network weights obtained from cross-validation training"
• [methods] We normalized the values in each feature attribution score matrix Si by dividing the significant threshold score identified from the background and clipped values to be between -1 and 1: "We normalized the values in each feature attribution score matrix Si by dividing the significant threshold score identified from the background and clipped values to be between -1 and 1"
• [methods] a threshold was set at the 97.5 percentile. Any feature attribution score in the observed dataset with an absolute value above the threshold was considered significant: "a threshold was set at the 97.5 percentile. Any feature attribution score in the observed dataset with an absolute value above the threshold was considered significant"
• [results] We identified 163 microbes that had at least one significant attribution score with a well-predicted metabolite: "We identified 163 microbes that had at least one significant attribution score with a well-predicted metabolite"
• [methods] the rows representing non-significant microbes were filtered out from S* as well as from all feature attribution score matrices S used in subsequent analyses: "the rows representing non-significant microbes were filtered out from S* as well as from all feature attribution score matrices S used in subsequent analyses"