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diff --git a/src/personalization/user_model/features.py b/src/personalization/user_model/features.py
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+++ b/src/personalization/user_model/features.py
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+import numpy as np
+from dataclasses import dataclass
+from sklearn.decomposition import PCA
+
+@dataclass
+class ItemProjection:
+    P: np.ndarray     # [k, d]
+    mean: np.ndarray  # [d]
+
+    @classmethod
+    def from_pca(cls, embeddings: np.ndarray, k: int) -> "ItemProjection":
+        """
+        embeddings: [M, d]
+        """
+        mean = embeddings.mean(axis=0)
+        centered = embeddings - mean
+        
+        # Ensure k is not larger than min(n_samples, n_features)
+        n_samples, n_features = embeddings.shape
+        actual_k = min(k, n_samples, n_features)
+        
+        pca = PCA(n_components=actual_k)
+        pca.fit(centered)
+
+        # pca.components_: [k, d]
+        P = pca.components_  # Each row is a principal component vector
+        
+        # If we had to reduce k, we might want to pad P or handle it?
+        # For now, let's assume we get what we asked for or less if data is small.
+        # But for the system we want fixed k. 
+        # If actual_k < k, we should pad with zeros to match expected dimension.
+        if actual_k < k:
+            padding = np.zeros((k - actual_k, n_features), dtype=P.dtype)
+            P = np.vstack([P, padding])
+            
+        return cls(P=P, mean=mean)
+
+    def transform_embeddings(self, E: np.ndarray) -> np.ndarray:
+        """
+        E: [N, d] -> [N, k]
+        """
+        return (E - self.mean) @ self.P.T
+
+    def transform_vector(self, e: np.ndarray) -> np.ndarray:
+        """
+        e: [d] -> [k]
+        """
+        return self.P @ (e - self.mean)
+