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Energy Manifold Natural Gradient Descent: From Riemannian Optimization to Modern Neuroscience, NeuroAI and Quantum Physics

Dr. Rishabh Pathak
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...
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Knowing Your Task and Knowing Your Data

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Dr. Rishabh Pathak

Before building a machine learning model, you must clearly understand the problem or task you want to solve. This means identifying:

  • The Goal: What question do you want to answer? For example, do you want to classify emails as spam or not spam? Detect fraudulent transactions? Or cluster customers based on purchasing behavior?
  • Supervised vs. Unsupervised: Determine whether your task is supervised (with labeled input-output pairs) or unsupervised (finding structure in unlabeled data).
  • Type of Prediction:
  • Classification: Predict a discrete label (e.g., species of an iris flower, type of fraud).
  • Regression: Predict a continuous value (e.g., house prices).
  • Ranking or Recommendations: Ordering items by relevance or suggesting products.

Understanding the task shapes your choices regarding which algorithms to use, how to evaluate success, and what features will be necessary.

Knowing Your Data

A deep knowledge of your data is equally important because:

  • Data Quality and Relevance: The features (attributes) should be relevant to the task. For example, having a patient's last name alone won’t help predict gender, but including the first name might, because some first names are gender-specific.
  • Feature Representation: How you represent your data usually has a larger impact on model performance than the precise choice of algorithm parameters.
  • Data Limitations: Knowing what information your data contains and what it does not is critical. Machine learning algorithms can't predict targets if the necessary information isn't there.
  • Distribution and Variability: Understanding how your data is distributed, if there are missing values, or if some classes are underrepresented will affect preprocessing, training, and model performance.

Practical Advice:

  • Don’t randomly throw data at algorithms without understanding the problem and data characteristics.
  • Ask key questions continuously during the project, such as:
    • What kind of data do I have?
    • What relationship do I expect between the input variables and the output?
    • What assumptions does my chosen algorithm make about the data?
    • Remember that the success of machine learning strongly depends on aligning your data and task understanding with an appropriate approach.

Summary

Knowing your task and knowing your data are foundational steps essential to designing an effective machine learning solution. Without this understanding, the performance of your model will suffer, and the insights gained may be misleading or irrelevant.

 

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