Complex matrix interpolation techniques from multi-manifold learning can be integrated with single-cell transcriptomics analysis to uncover hidden structural patterns in Multiple Sclerosis disease progression.

Multi-manifold learning matrix interpolation methods (specifically, techniques such as geodesic interpolation on Grassmann/Stiefel manifolds or coupled manifold alignment) applied to single-cell RNA sequencing (scRNA-seq) data from Multiple Sclerosis (MS) patient cohorts will reveal statistically significant latent structural patterns in disease progression that are not detectable by standard dimensionality reduction methods (PCA, UMAP, t-SNE alone), as measured by: (1) improved cluster separation (silhouette score ≥ 0.15 above baseline), (2) identification of ≥2 novel cell-state transition trajectories validated by orthogonal marker gene expression, and (3) significant correlation (r ≥ 0.40, p < 0.05) between interpolated manifold coordinates and clinical MS progression scores (EDSS or MSSS).

PHASE 1 — Data Preparation and Baseline (Days 1–14): Acquire 3 publicly available MS scRNA-seq datasets. Apply standard QC (doublet removal via Scrublet, mitochondrial gene filtering <20%, minimum 200 genes/cell). Normalize (scran pooling normalization), select 3,000 highly variable genes, apply Harmony batch correction. Compute baseline dimensionality reductions: PCA (50 PCs), UMAP (n_neighbors=15, min_dist=0.1), t-SNE (perplexity=30). Cluster with Leiden algorithm (resolution=0.5). Record baseline silhouette scores, cluster purity, and trajectory inference (PAGA) results.

PHASE 2 — Multi-Manifold Implementation (Days 15–35): Implement 3 matrix interpolation strategies: (A) Grassmann manifold interpolation on PCA subspace matrices, (B) coupled NMF with manifold-regularized interpolation, (C) diffusion map-based multi-condition interpolation. For each method, interpolate between disease-state manifolds at 5 interpolation steps. Extract interpolated coordinates and latent factors.

PHASE 3 — Structural Pattern Analysis (Days 36–50): Apply trajectory inference (Monocle3, scVelo) to interpolated embeddings. Identify novel cell states by differential expression (DESeq2, FDR <0.05, |log2FC| >1.5). Validate novel states against published marker databases (CellMarker 2.0, PanglaoDB). Correlate manifold coordinates with clinical metadata.

PHASE 4 — Statistical Validation (Days 51–60): Bootstrap resampling (n=500) for confidence intervals. Permutation testing for clinical correlations. Cross-dataset replication. Comparison against 4 baseline methods.

# ============================================================
# Multi-Manifold scRNA-seq MS Analysis Pipeline
# Architecture: 4-stage modular pipeline
# ============================================================

# --- STAGE 1: DATA INGESTION & QC ---
import scanpy as sc
import scvi
import harmony
import scrublet as scr
import numpy as np
import geomstats.geometry.grassmannian as grassmann
from geomstats.geometry.grassmannian import Grassmannian
from pymanopt.manifolds import Grassmann as PyGrassmann
import pandas as pd
from scipy import stats

def load_and_qc(geo_ids: list, min_genes=200, max_genes=6000, max_mito=0.20):
    """Load GEO datasets and apply QC filters."""
    adatas = {}
    for geo_id in geo_ids:
        adata = sc.read_10x_h5(f"data/{geo_id}/filtered_feature_bc_matrix.h5")
        # Doublet detection
        scrub = scr.Scrublet(adata.X)
        doublet_scores, predicted_doublets = scrub.scrub_doublets(threshold=0.25)
        adata.obs['doublet_score'] = doublet_scores
        adata.obs['predicted_doublet'] = predicted_doublets
        # Mitochondrial filtering
        adata.var['mt'] = adata.var_names.str.startswith('MT-')
        sc.pp.calculate_qc_metrics(adata, qc_vars=['mt'], inplace=True)
        # Apply filters
        sc.pp.filter_cells(adata, min_genes=min_genes)
        sc.pp.filter_cells(adata, max_genes=max_genes)
        adata = adata[adata.obs.pct_counts_mt < max_mito * 100]
        adata = adata[~adata.obs.predicted_doublet]
        adatas[geo_id] = adata
    return adatas

# --- STAGE 2: NORMALIZATION, HVG, BATCH CORRECTION ---
def preprocess_and_correct(adatas: dict, n_hvg=3000, n_pcs=50):
    """Normalize, select HVGs, and apply Harmony batch correction."""
    # Concatenate datasets
    adata_combined = sc.concat(adatas, label='dataset', keys=list(adatas.keys()))
    # Normalization
    sc.pp.normalize_total(adata_combined, target_sum=1e4)
    sc.pp.log1p(adata_combined)
    # HVG selection
    sc.pp.highly_variable_genes(adata_combined, n_top_genes=n_hvg, 
                                 flavor='seurat_v3', batch_key='dataset')
    adata_combined = adata_combined[:, adata_combined.var.highly_variable]
    # PCA
    sc.pp.scale(adata_combined, max_value=10)
    sc.tl.pca(adata_combined, n_comps=n_pcs, svd_solver='arpack')
    # Harmony batch correction
    import harmonypy as hm
    ho = hm.run_harmony(adata_combined.obsm['X_pca'], 
                        adata_combined.obs, 
                        vars_use=['dataset', 'donor_id'],
                        theta=[2, 1], lambda_val=1)
    adata_combined.obsm['X_pca_harmony'] = ho.Z_corr.T
    return adata_combined

# --- STAGE 3: MULTI-MANIFOLD INTERPOLATION ---

class GrassmannInterpolator:
    """
    Interpolates between disease-condition PCA subspaces 
    on the Grassmann manifold.
    """
    def __init__(self, n_components=50, n_interpolation_steps=5):
        self.k = n_components
        self.n_steps = n_interpolation_steps
        self.manifold = Grassmannian(n=3000, k=n_components)  # Gr(k, n)
        
    def fit_condition_subspaces(self, adata, condition_key='disease_state'):
        """Compute PCA subspace matrix for each condition."""
        self.subspaces = {}
        conditions = adata.obs[condition_key].unique()
        for cond in conditions:
            mask = adata.obs[condition_key] == cond
            X_cond = adata[mask].obsm['X_pca_harmony']  # (n_cells, k)
            # Orthonormal basis via QR decomposition
            Q, _ = np.linalg.qr(X_cond.T)  # (n_features, k)
            self.subspaces[cond] = Q[:self.k, :self.k]  # Point on Gr(k,n)
        return self
    
    def interpolate(self, cond_start, cond_end):
        """Compute geodesic interpolation between two condition subspaces."""
        U_start = self.subspaces[cond_start]
        U_end = self.subspaces[cond_end]
        # Geodesic path on Grassmann manifold
        # Using logarithmic map + linear interpolation + exponential map
        interpolated_subspaces = []
        for t in np.linspace(0, 1, self.n_steps):
            # Log map at U_start
            log_vec = self.manifold.metric.log(U_end, U_start)
            # Scale by t
            scaled_vec = t * log_vec
            # Exp map back to manifold
            U_t = self.manifold.metric.exp(scaled_vec, U_start)
            interpolated_subspaces.append(U_t)
        return interpolated_subspaces
    
    def project_cells(self, adata, interpolated_subspaces):
        """Project all cells onto each interpolated subspace."""
        X = adata.obsm['X_pca_harmony']
        projections = []
        for U_t in interpolated_subspaces:
            # Project: X_proj = X @ U_t @ U_t.T (reconstruction in subspace)
            X_proj = X @ U_t.T  # (n_cells, k)
            projections.append(X_proj)
        return np.stack(projections, axis=0)  # (n_steps, n_cells, k)


class CoupledNMFInterpolator:
    """
    Coupled NMF with manifold regularization for cross-condition interpolation.
    """
    def __init__(self, rank=30, alpha_reg=0.1, max_iter=500):
        self.rank = rank
        self.alpha = alpha_reg
        self.max_iter = max_iter
    
    def fit_and_interpolate(self, adata, condition_key='disease_state'):
        from sklearn.decomposition import NMF
        from sklearn.neighbors import kneighbors_graph
        from scipy.sparse.csgraph import laplacian
        
        conditions = sorted(adata.obs[condition_key].unique())
        W_matrices = {}
        H_matrices = {}
        
        for cond in conditions:
            mask = adata.obs[condition_key] == cond
            X_cond = np.abs(adata[mask].X.toarray() if hasattr(adata[mask].X, 'toarray') 
                           else adata[mask].X)
            # Build cell graph for manifold regularization
            G = kneighbors_graph(X_cond, n_neighbors=15, mode='connectivity')
            L = laplacian(G, normed=True)
            # NMF with graph regularization (alternating updates)
            model = NMF(n_components=self.rank, beta_loss='kullback-leibler',
                       solver='mu', max_iter=self.max_iter, init='nndsvda')
            W = model.fit_transform(X_cond)
            H = model.components_
            # Manifold regularization: add alpha * Tr(W.T @ L @ W) penalty
            # (simplified: post-hoc smoothing via graph diffusion)
            W_smooth = np.linalg.solve(np.eye(W.shape[0]) + self.alpha * L.toarray(), W)
            W_matrices[cond] = W_smooth
            H_matrices[cond] = H
        
        # Interpolate W matrices between conditions
        interpolated = {}
        cond_pairs = [(conditions[i], conditions[i+1]) for i in range(len(conditions)-1)]
        for c1, c2 in cond_pairs:
            steps = []
            for t in np.linspace(0, 1, 5):
                # Geodesic interpolation on positive orthant (NMF constraint)
                W_t = (1 - t) * W_matrices[c1] + t * W_matrices[c2]
                W_t = np.maximum(W_t, 0)  # Enforce non-negativity
                steps.append(W_t)
            interpolated[(c1, c2)] = steps
        return interpolated, W_matrices, H_matrices


# --- STAGE 4: DOWNSTREAM ANALYSIS ---

def compute_silhouette_comparison(adata, embedding_keys: list, label_key='cell_type'):
    """Compare silhouette scores across embedding methods."""
    from sklearn.metrics import silhouette_score
    results = {}
    labels = adata.obs[label_key].values
    for key in embedding_keys:
        X_embed = adata.obsm[key]
        score = silhouette_score(X_embed, labels, metric='euclidean', 
                                  sample_size=min(10000, len(labels)))
        results[key] = score
    return results

def identify_novel_cell_states(adata, interpolated_embedding, 
                                 baseline_clusters, resolution=0.5):
    """Find clusters in interpolated space absent from baseline."""
    from sklearn.metrics import jaccard_score
    import scanpy as sc
    
    adata.obsm['X_interpolated'] = interpolated_embedding
    sc.pp.neighbors(adata, use_rep='X_interpolated', n_neighbors=15)
    sc.tl.leiden(adata, resolution=resolution, key_added='leiden_interpolated')
    
    novel_clusters = []
    for new_clust in adata.obs['leiden_interpolated'].unique():
        new_mask = (adata.obs['leiden_interpolated'] == new_clust).values
        max_jaccard = 0
        for base_clust in baseline_clusters:
            base_mask = (adata.obs['leiden_baseline'] == base_clust).values
            j = jaccard_score(new_mask, base_mask)
            max_jaccard = max(max_jaccard, j)
        if max_jaccard < 0.30:  # Novel if <30% overlap with any baseline cluster
            novel_clusters.append(new_clust)
    return novel_clusters

def clinical_correlation_analysis(adata, manifold_coords, clinical_df, 
                                    clinical_col='EDSS'):
    """Correlate per-donor manifold coordinates with clinical scores."""
    donors = adata.obs['donor_id'].unique()
    donor_coords = []
    donor_scores = []
    
    for donor in donors:
        if donor in clinical_df.index:
            mask = adata.obs['donor_id'] == donor
            mean_coord = manifold_coords[mask].mean(axis=0)
            donor_coords.append(mean_coord)
            donor_scores.append(clinical_df.loc[donor, clinical_col])
    
    donor_coords = np.array(donor_coords)
    donor_scores = np.array(donor_scores)
    
    # Pearson correlation for each manifold dimension
    correlations = []
    for dim in range(donor_coords.shape[1]):
        r, p = stats.pearsonr(donor_coords[:, dim], donor_scores)
        correlations.append({'dim': dim, 'r': r, 'p': p})
    
    # Also compute correlation with first PC of manifold coords
    from sklearn.decomposition import PCA
    pca = PCA(n_components=1)
    pc1 = pca.