Projecting Data and Reconstructing from Components
Learners will transform a dataset to the principal-component space, visualise the 2D projection, and reconstruct the original features to quantify information loss.
Projecting Data and Reconstructing from Components is a free Machine Learning Academy lesson on CoddyKit — lesson 2 of 4. You can read the complete lesson below for free — then practise it hands-on in the browser with a built-in code editor and a 24/7 AI tutor. It is part of the Machine Learning Academy learning path, one of 4 lessons in the course, and your progress syncs across the web and the CoddyKit app.
Projection: From High-D to Low-D
After PCA finds the principal components, projection transforms each data point into the new component space. The projected coordinates are called scores. If you keep only 2 components from 64 original features, each 64-dimensional point becomes a 2-dimensional score. This is achieved by multiplying the centred data matrix by the matrix of eigenvectors (the loadings matrix).
The transform Method in sklearn
In scikit-learn, pca.fit(X) learns the components and pca.transform(X) projects the data. The convenience method pca.fit_transform(X) does both in one call. The result is a matrix of shape (n_samples, n_components) — each row is a point in the reduced space.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
X, y = load_digits(return_X_y=True) # 1797 x 64
X_scaled = StandardScaler().fit_transform(X)
pca = PCA(n_components=10)
X_reduced = pca.fit_transform(X_scaled)
print('Original shape:', X_scaled.shape)
print('Reduced shape:', X_reduced.shape)
print('Variance retained:', pca.explained_variance_ratio_.sum().round(4))Visualising the 2D Projection
Projecting to 2 components gives a scatter plot where class separation is often visible even though labels were never used during PCA. This is an important exploratory tool: if classes are well-separated in 2D PCA space, a simple linear classifier may perform well in the full-dimensional space.
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
X, y = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
pca2 = PCA(n_components=2)
X_2d = pca2.fit_transform(X_scaled)
plt.figure(figsize=(8, 6))
for digit in range(10):
mask = y == digit
plt.scatter(X_2d[mask, 0], X_2d[mask, 1], label=str(digit), s=10, alpha=0.6)
plt.legend(title='Digit', bbox_to_anchor=(1, 1))
plt.title('MNIST digits in 2D PCA space')
plt.tight_layout()
plt.show()Reconstruction: Going Back to Original Space
Reconstruction reverses the projection: multiply the reduced scores by the transpose of the loadings matrix and add back the mean. The result is an approximation of the original data in the original feature space. Perfect reconstruction is only possible if you kept all components; retaining fewer introduces reconstruction error.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
import numpy as np
X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
pca = PCA(n_components=20)
X_reduced = pca.fit_transform(X_scaled)
# Reconstruct back to 64 dimensions
X_reconstructed = pca.inverse_transform(X_reduced)
print('Reconstruction shape:', X_reconstructed.shape)
# Mean squared reconstruction error
mse = np.mean((X_scaled - X_reconstructed) ** 2)
print(f'MSE: {mse:.4f}')Visualising Reconstruction Quality
For image data, you can plot original and reconstructed images side by side. With more components retained, the reconstruction looks sharper. With very few components, digits become blurry blobs. This visual comparison is a powerful communication tool for showing stakeholders the trade-off between compression and information loss.
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
fig, axes = plt.subplots(3, 5, figsize=(12, 7))
component_counts = [1, 2, 5, 10, 30]
for col, nc in enumerate(component_counts):
pca = PCA(n_components=nc)
X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
# Un-standardise for display (approximate)
axes[0, col].imshow(X[0].reshape(8, 8), cmap='gray')
axes[0, col].set_title(f'Original' if col == 0 else '')
axes[1, col].imshow(X_r[0].reshape(8, 8), cmap='gray')
axes[1, col].set_title(f'n={nc}')
plt.tight_layout()
plt.show()Reconstruction Error vs Number of Components
Plot reconstruction MSE against the number of components to see the information-loss curve. This is the quantitative version of the visual comparison. A sharp decrease in MSE as you add the first few components mirrors the scree plot, confirming that most information lives in a small subspace.
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
import numpy as np
X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
components = [1, 2, 5, 10, 20, 30, 40, 50, 64]
mse_values = []
for nc in components:
pca = PCA(n_components=nc)
X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
mse_values.append(np.mean((X_scaled - X_r) ** 2))
plt.plot(components, mse_values, marker='o')
plt.xlabel('Number of components')
plt.ylabel('Reconstruction MSE')
plt.title('Information Loss vs Compression')
plt.show()Interpretting Reconstruction Error
At zero components, reconstruction error equals the total variance of the data. At full components, error is zero. The ratio 1 - explained_variance_ratio.sum() tells you the fraction of variance discarded. For most practical ML pipelines, keeping 95–99% of variance (and discarding 1–5%) loses very little predictive signal while significantly reducing feature count and training time.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
for nc in [5, 10, 20, 30, 40]:
pca = PCA(n_components=nc)
pca.fit(X_scaled)
retained = pca.explained_variance_ratio_.sum()
print(f'n_components={nc:2d} retained={retained:.3f} discarded={1-retained:.3f}')Using inverse_transform in Practice
pca.inverse_transform(X_reduced) is a method on the fitted PCA object. It returns the data in the original feature space but with the information from discarded components zeroed out. This is useful for anomaly detection: reconstruct training data and flag points with high reconstruction error as outliers that the PCA model could not represent well.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import numpy as np
# Simulated normal vs anomalous points
X_normal = np.random.randn(100, 10)
X_anomaly = np.random.randn(5, 10) * 10 # far from origin
X_all = np.vstack([X_normal, X_anomaly])
X_scaled = StandardScaler().fit_transform(X_all)
pca = PCA(n_components=5)
X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
errors = np.mean((X_scaled - X_r) ** 2, axis=1)
print('Max error index:', np.argmax(errors), '(anomalies start at index 100)')Whitening: Decorrelated Components with Unit Variance
Setting PCA(whiten=True) scales the projected scores so each component has unit variance. This removes correlations between components and can improve the performance of algorithms like SVMs or neural networks that are sensitive to feature scale. Whitening is standard preprocessing before training on PCA-reduced features.
from sklearn.decomposition import PCA
from sklearn.datasets import load_iris
from sklearn.preprocessing import StandardScaler
import numpy as np
X, _ = load_iris(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)
pca_white = PCA(n_components=3, whiten=True)
X_w = pca_white.fit_transform(X_scaled)
print('Component variances (should be 1.0):', np.var(X_w, axis=0).round(4))PCA Limitations on Non-Linear Data
PCA finds only linear projections. If data lies on a curved surface — like a Swiss roll — PCA projects onto a flat plane, destroying the manifold structure. In such cases, consider Kernel PCA with an RBF kernel or non-linear alternatives like t-SNE or UMAP for exploration. For model preprocessing, however, linear PCA is usually sufficient and much faster.
Project and Reconstruct: Complete Workflow
A clean PCA pipeline always follows the same pattern: standardise, fit PCA on training data, transform train and test separately, optionally reconstruct to inspect quality.
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_digits
import numpy as np
X, y = load_digits(return_X_y=True)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)
scaler = StandardScaler()
X_train_s = scaler.fit_transform(X_train)
X_test_s = scaler.transform(X_test) # use train scaler
pca = PCA(n_components=0.95)
X_train_r = pca.fit_transform(X_train_s) # fit only on train
X_test_r = pca.transform(X_test_s) # transform test
print(f'Reduced: {X_train_r.shape[1]} components from 64 features')Quick Check
Test your understanding of PCA projection and reconstruction from this lesson.
Lesson Recap
In this lesson you learned: pca.transform projects data into component space with shape (n_samples, n_components), pca.inverse_transform reconstructs data in original feature space with information from discarded components lost, and reconstruction error quantifies information loss and can flag anomalies. Next up we explore t-SNE — a non-linear technique for 2D visualisation of high-dimensional data.
Frequently asked questions
Is the “Projecting Data and Reconstructing from Components” lesson free?
Yes — the full text of “Projecting Data and Reconstructing from Components” is free to read here on the web, and the Machine Learning Academy course includes 4 lessons in total. To practise it interactively (a built-in code editor and a 24/7 AI tutor) and unlock the rest of the Machine Learning Academy course, upgrade to CoddyKit PRO.
What will I learn in “Projecting Data and Reconstructing from Components”?
Learners will transform a dataset to the principal-component space, visualise the 2D projection, and reconstruct the original features to quantify information loss. You practise Machine Learning Academy with hands-on code you run directly in the browser, and a 24/7 AI tutor answers your questions as you work through the lesson.
Do I need any experience to start Machine Learning Academy?
No prior experience is required. Machine Learning Academy on CoddyKit is structured for beginners through advanced learners; this is — lesson 2 of 4, so you can start here or from the beginning and move at your own pace.
How long does the “Projecting Data and Reconstructing from Components” lesson take?
Most CoddyKit lessons take about 5–10 minutes. Each one is bite-sized and interactive, so you make steady progress and pick up exactly where you left off across the web and the app.
Can I write and run code in this Machine Learning Academy lesson?
Yes. Every Machine Learning Academy lesson includes a built-in code editor, so you write and run real code right in your browser and get instant AI feedback — no local setup required.
All lessons in this course
- PCA: Variance, Eigenvectors, and Principal Components
- Projecting Data and Reconstructing from Components
- t-SNE: Neighbourhood Preservation for Visualisation
- PCA as Preprocessing: Speed and Noise Reduction in Pipelines