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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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Judgement Sampling

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Dr. Rishabh Pathak

Judgment sampling, also known as purposive or selective sampling, is a non-probability sampling technique where researchers use their judgment and expertise to select sample units based on specific criteria or characteristics relevant to the research objectives. In judgment sampling, researchers intentionally choose sample units that they believe are representative or typical of the population of interest. Here are some key points about judgment sampling:


1.    Definition:

§  Judgment sampling is a non-probability sampling method where researchers select sample units based on their judgment, expertise, or knowledge of the population.

§  Sample units are chosen deliberately to represent certain traits, characteristics, or experiences that are deemed relevant to the research objectives.

2.    Characteristics:

§  Judgment sampling relies on the researcher's subjective judgment and understanding of the population to select sample units that are considered typical, informative, or representative.

§  Researchers may use their expertise to identify key characteristics or criteria for selecting sample units that align with the research focus.

3.    Types of Judgment Sampling:

§  Convenience Sampling: Selecting sample units based on their accessibility, availability, or convenience to the researcher.

§  Expert Sampling: Choosing sample units based on the expertise, knowledge, or qualifications of the individuals selected.

§  Typical Case Sampling: Selecting sample units that are considered typical or illustrative of the population's characteristics or behaviors.

4.    Advantages:

§  Judgment sampling allows researchers to focus on specific characteristics or traits of interest, making it suitable for targeted research objectives or exploratory studies.

§  This method is valuable for qualitative research, case studies, and situations where in-depth insights or unique perspectives are sought.

5.    Limitations:

§  Results obtained from judgment samples may be subject to bias, as the selection of sample units is based on the researcher's subjective judgment rather than randomization.

§  The generalizability of findings from judgment sampling may be limited, as the sample may not be representative of the entire population.

6.    Applications:

§  Judgment sampling is commonly used in qualitative research, ethnographic studies, and exploratory research where researchers seek to understand specific phenomena or behaviors.

§  This method is particularly useful when studying unique populations, rare events, or complex phenomena that require expert judgment in sample selection.

7.    Considerations:

§  Researchers should clearly define the criteria for selecting sample units in judgment sampling and justify their choices based on the research objectives.

§  While judgment sampling offers flexibility and targeted sampling, researchers should acknowledge its limitations in terms of generalizability and potential bias.

Judgment sampling is a valuable sampling technique that allows researchers to strategically select sample units based on specific criteria or characteristics relevant to their research goals. While this method offers advantages in terms of targeted sampling and in-depth exploration, researchers should be mindful of its limitations in terms of representativeness and potential bias. Careful consideration of the research objectives and criteria for sample selection is essential when employing judgment sampling in a study.

 

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