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The Misaligned Projection

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🐛
The Misaligned Projection
Started immediately after the PCA update last Friday.

Dimensionality reduction for a recommendation system processing 500K user-item interactions daily.

Symptom
The model's recommendation accuracy dropped from 0.85 to 0.65 after the latest PCA update.
📡
Signals
metric
Recommendation accuracy on the test set
Dropped from 0.85 to 0.65 after PCA update
log
PCA transformation logs
Eigenvalues contain negative values
distribution
Distribution of projected data
Projected data has a mean of 0.5, expected mean was 0
code
PCA implementation code review
Covariance matrix calculation uses incorrect normalization
🔬
Hypotheses
Covariance matrix is incorrectly computed, leading to wrong eigenvectors.
Supporting Signals
Eigenvalues contain negative valuesCovariance matrix calculation uses incorrect normalization
Verify: Check the covariance matrix computation for correct normalization.
cov_matrix = np.cov(data, rowvar=False); print(np.all(np.linalg.eigvals(cov_matrix) >= 0))
Projection step is misaligned due to incorrect eigenvectors.
Supporting Signals
Projected data has a mean of 0.5, expected mean was 0
Verify: Verify the eigenvectors used for projection are correct.
eigvals, eigvecs = np.linalg.eigh(cov_matrix); print(np.dot(eigvecs.T, eigvecs))
🎯
Root Cause
Covariance matrix was computed without centering the data, leading to incorrect eigenvectors.
Why It Happened

A recent code change removed the data centering step before covariance calculation.

🔧
Fix Strategy
ImmediateRollback to the previous PCA implementation.
Short-termReintroduce data centering before computing the covariance matrix.
Long-termImplement a validation step to ensure eigenvalues are non-negative after PCA computation.
🐍
Python Fix
Python
data_centered = data - np.mean(data, axis=0); cov_matrix = np.cov(data_centered, rowvar=False); eigvals, eigvecs = np.linalg.eigh(cov_matrix)
✗
Common Wrong Diagnoses
▼ Show
The PCA implementation is correct, but the data distribution changed.
Why candidates think this: The drop in accuracy could be attributed to a shift in data distribution.
Why wrong: The issue started immediately after the PCA update, not due to data changes.
The eigenvectors are computed correctly, but the projection step is flawed.
Why candidates think this: The mean shift in projected data suggests a projection issue.
Why wrong: The projection issue is a consequence of incorrect eigenvectors, not the cause.
📊
Monitoring to Add
  • Check for non-negative eigenvalues after PCA computation.
  • Monitor the mean of projected data to ensure it remains centered.
  • Log changes in covariance matrix computation for review.
📋
Process Change

Require code reviews for all changes to core mathematical operations.

🏗️
Architectural Change

Introduce a PCA validation module that checks for common errors like data centering.

💡
Key Insight
Always center data before computing the covariance matrix in PCA to ensure correct eigenvector computation.

Practice

(1/5)
1. 
import numpy as np

def cosine_similarity(a, b):
    dot_product = np.dot(a, b)
    norm_a = np.linalg.norm(a)
    norm_b = np.linalg.norm(b)
    return dot_product / (norm_a * norm_b)

vector1 = np.array([1, 2, 3])
vector2 = np.array([4, 5, 6])

result = cosine_similarity(vector1, vector2)
print(result)
easy
A. 0.9746318461970762
B. 0.9999999999999998
C. 1.0
D. 0.0

Solution

  1. Step 1: Calculate dot product and norms

    Dot product = 32, norm_a = 3.7416573867739413, norm_b = 8.774964387392123.

  2. Step 2: Compute cosine similarity

    Cosine similarity = 32 / (3.7416573867739413 * 8.774964387392123) = 0.9746318461970762.

  3. Final Answer:

    Option A -> Option A

  4. Quick Check:

    Dot product over product of norms. [OK]
Hint: Dot product over product of norms. [OK]
Common Mistakes:
2. 
import numpy as np

data = np.array([1, 2, 3, 4, 5])
mean = np.mean(data)
std = np.std(data)

standardized_data = (data - mean) / std
print(standardized_data)
What does the code output?
easy
A. An array with values [0, 1, 2, 3, 4]
B. An array with values [0, 0.5, 1, 1.5, 2]
C. An array with values [-1.41, -0.71, 0, 0.71, 1.41]
D. An array with values [-2, -1, 0, 1, 2]

Solution

  1. Step 1: Understand Standardization

    Standardization transforms data to have a mean of 0 and a standard deviation of 1.

  2. Step 2: Calculate the Output

    The standardized values are calculated as (data - mean) / std, resulting in [-1.41, -0.71, 0, 0.71, 1.41].

  3. Final Answer:

    Option C -> Option C

  4. Quick Check:

    Standardization centers data around 0. [OK]
Hint: Standardization centers data around 0. [OK]
Common Mistakes:
3. What does the following Python code snippet output when implementing sinusoidal positional encoding?
import numpy as np

def positional_encoding(position, d_model):
    angle_rads = np.arange(position)[:, np.newaxis] / np.power(10000, (2 * (np.arange(d_model)[np.newaxis, :] // 2)) / np.float32(d_model))
    angle_rads[:, 0::2] = np.sin(angle_rads[:, 0::2])
    angle_rads[:, 1::2] = np.cos(angle_rads[:, 1::2])
    return angle_rads

pe = positional_encoding(4, 4)
print(pe)
easy
A. A 4x4 matrix with alternating sine and cosine values
B. A 4x4 matrix with random values
C. A 4x4 identity matrix
D. A 4x4 matrix with all zeros

Solution

  1. Step 1: Understand the code logic

    The code generates a matrix where each row corresponds to a position and each column to a dimension, filled with sinusoidal values.

  2. Step 2: Eliminate wrong options

    A 4x4 matrix with all zeros is incorrect because the matrix is not filled with zeros. A 4x4 identity matrix is incorrect because the matrix is not an identity matrix. A 4x4 matrix with random values is incorrect because the values are not random but deterministic sinusoidal values.

  3. Final Answer:

    Option A -> Option A

  4. Quick Check:

    Sinusoidal encoding alternates sine/cosine. [OK]
Hint: Sinusoidal encoding alternates sine/cosine. [OK]
Common Mistakes:
4. 
import numpy as np

confusion_matrix = np.array([[50, 10], [5, 35]])

precision = confusion_matrix[0, 0] / (confusion_matrix[0, 0] + confusion_matrix[1, 0])
recall = confusion_matrix[0, 0] / (confusion_matrix[0, 0] + confusion_matrix[0, 1])

print(f"Precision: {precision}, Recall: {recall}")
easy
A. Precision: 0.8333, Recall: 0.8333
B. Precision: 0.9091, Recall: 0.9091
C. Precision: 0.8333, Recall: 0.9091
D. Precision: 0.9091, Recall: 0.8333

Solution

  1. Step 1: Calculate Precision

    Precision is calculated as TP / (TP + FP) = 50 / (50 + 5) = 0.9091.

  2. Step 2: Calculate Recall

    Recall is calculated as TP / (TP + FN) = 50 / (50 + 10) = 0.8333.

  3. Final Answer:

    Option D -> Option D

  4. Quick Check:

    Precision uses FP, Recall uses FN. [OK]
Hint: Precision uses FP, Recall uses FN. [OK]
Common Mistakes:
5. A logistic regression model is not converging during training. Which line in the code is most likely causing the issue?
import numpy as np

# Initialize weights
weights = np.zeros((m, 1))

# Training loop
for epoch in range(epochs):
    # Compute predictions
    predictions = 1 / (1 + np.exp(-np.dot(X, weights)))
    
    # Compute gradient
    gradient = np.dot(X.T, (predictions - y)) / n
    
    # Update weights
    weights -= learning_rate * gradient
medium
A. weights -= learning_rate * gradient
B. predictions = 1 / (1 + np.exp(-np.dot(X, weights)))
C. weights = np.zeros((m, 1))
D. gradient = np.dot(X.T, (predictions - y)) / n

Solution

  1. Step 1: Identify potential issues

    Convergence issues often arise from incorrect weight updates or learning rate problems.

  2. Step 2: Analyze weight update line

    If learning_rate is too high, weights update too aggressively, causing divergence.

  3. Final Answer:

    Option A -> Option A

  4. Quick Check:

    Check learning rate in weight updates [OK]
Hint: Check learning rate in weight updates [OK]
Common Mistakes:
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