When Geometry, Energy, Artificial Intelligence and Neuroscience Converge Modern Artificial Intelligence is rapidly moving beyond the idea that learning simply means minimizing an error function. Increasingly, researchers are asking a deeper question: what is the structure of the space in which learning takes place? This question becomes particularly important when the system being modelled is constrained, nonlinear, dynamic, or governed by physical principles. A recent work titled “Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers” , by Zhangyong Liang and Huanhuan Gao, introduces Energy Manifold Natural Gradient Descent (EMNGD) , a mathematical framework that extends energy-based natural-gradient optimization from unconstrained Euclidean parameter spaces to constrained Riemannian parameter manifolds . At its core, the framework proposes a simple but powerful principle: An optimization algorithm should not only determine how to reduce error; it sh...
1. What is the Decision Function?
- The decision_function
method is provided by many classifiers in scikit-learn.
- It returns a continuous
score for each sample, representing the classifier’s
confidence or margin.
- This score reflects how strongly the model favors one class over another
in binary classification, or a more complex set of scores in multiclass
classification.
2. Shape and Output of
decision_function
- For binary
classification, the output shape is (n_samples,).
- Each value is a floating-point number indicating the degree
to which the sample belongs to the positive class.
- Positive values indicate a preference for the positive
class; negative values indicate a preference for the negative class.
- For multiclass
classification, the output is usually a 2D array of shape
(n_samples, n_classes), providing scores for each class.
3. Interpretation of
decision_function Scores
- The sign of the value (positive or negative) determines
the predicted class.
- The magnitude
represents the confidence or "distance" from the decision boundary.
- The larger the absolute value, the more confident the
model is in its classification.
Example:
print("Decision function values:\n", classifier.decision_function(X_test)[:6])# Outputs something like:# [4.5, -1.2, 0.3, 5.0, -3.1, ...]- Here, values like 4.5 or 5.0 indicate strong confidence
in the positive class; -1.2 or -3.1 indicate strong preference for the
negative class.
4. Relationship to Prediction
Threshold
- For binary
classifiers, prediction is derived by thresholding:
- Predicted class = positive if decision_function score
> 0.
- Predicted class = negative otherwise.
- This threshold can be adjusted:
- Changing threshold impacts false positives/negatives.
- Adjusting threshold can improve metrics like precision
and recall in imbalanced data.
5. Examples of Classifiers Using
decision_function
- Support Vector Machines (SVMs) use decision_function to
provide margin distances from the decision boundary.
- GradientBoostingClassifier also provides
decision_function for more granular confidence.
- Logistic regression usually does not provide
decision_function but provides predict_proba instead (log odds can be
considered similar).
6. Advantages of
decision_function Over predict_proba
- decision_function outputs raw scores, which might be more
informative for some models.
- These raw scores can be transformed into probabilities
with calibration methods like Platt
scaling.
- For models like SVMs, predict_proba is a wrapper over
decision_function with a calibration step.
- Users can set custom thresholds on decision_function to
better control classification decisions.
7. Use in Model Evaluation
- decision_function outputs enable construction of ROC curves, which plot
True Positive Rate vs False Positive Rate at different thresholds.
- By varying the decision threshold, you can evaluate model
performance across thresholds.
- Thus, decision_function is crucial for comprehensive
model assessment beyond accuracy.
8. Example Code Snippet (from the
book)
from sklearn.ensemble import GradientBoostingClassifier # Suppose we have a trained GradientBoostingClassifier called gbrtprint("X_test.shape:", X_test.shape)print("Decision function shape:", gbrt.decision_function(X_test).shape) print("Decision function:\n", gbrt.decision_function(X_test)[:6])Output might be:
X_test.shape: (25, 2)Decision function shape: (25,)Decision function:[4. 2.5 1.3 0.7 -1.2 -3.4]Explanation: These values show
the strength of model preference for the positive class.
9. Summary Points
|
Aspect |
|
|
Purpose |
Measures
confidence or margin in classification |
|
Output
(Binary) |
Array
of floats (n_samples,) indicating class preference |
|
Output
(Multiclass) |
Array
of floats (n_samples, n_classes) with scores per class |
|
Interpretation |
Positive
= positive class, Negative = negative class; magnitude = confidence |
|
Thresholding |
Default
threshold at 0 to convert to class labels |
|
Usage |
Enables
custom thresholds, ROC analysis, model calibration |
|
Example
models |
SVM, Gradient
Boosting, some ensemble classifiers |
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