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...
The normal equations are a
mathematical formulation used in linear regression to find the best-fitting
line (or hyperplane) through a set of data points. They provide a way to
directly compute the parameters (coefficients) of a linear model.
1. Overview of Linear Regression
In linear regression, we aim to
model the relationship between a dependent variable y
and one or more independent variables (features) x1, x2,…, xp. The
model can be expressed in the following linear form:
y=θ0+θ1x1+θ2x2+…+θpxp
Where:
- θ₀ is
the intercept,
- θ1,…,θp are the coefficients for
the independent variables.
2. Objective of Linear Regression
The goal is to find the
coefficients θ (represented as a vector) such that the
predicted values y^ minimize the sum of the
squared differences between the observed values y and
the predicted values y^:
J(θ)=∑i=1n(y(i)−y^(i))2=∑i=1n(y(i)−θTx(i))2
Where x(i) is the
feature vector for the i-th observation, and y^(i)=θTx(i).
3. Deriving the Normal Equations
To minimize the cost function J(θ), we perform gradient descent or directly derive the
normal equations. The derivation involves taking the gradient of the cost
function and setting it to zero.
Step 1: Matrix Formulation
Let X be
the design matrix where each row corresponds to a training example and each
column corresponds to a feature:
X=11⋮1x11x21⋮xn1x12x22⋮xn2……⋱…x1px2p⋮xnp
The vector of outputs y can be represented as:
y=y(1)y(2)⋮y(n)
And the parameters can be
represented as a vector:
θ=θ0θ1⋮θp
Step 2: Cost Function in Matrix
Form
The cost function can now be
expressed in matrix form as:
J(θ)=(y−Xθ)T(y−Xθ)=yTy−2θTXTy+θTXTXθ
Step 3: Gradient Calculation
We take the gradient with respect
to θ:
∇J(θ)=−2XTy+2XTXθ
Step 4: Setting Gradient to Zero
Setting the gradient to zero for
minimization:
−2XTy+2XTXθ=0
This simplifies to:
XTXθ=XTy
This is the normal equation. If XTX is invertible, we can solve for θ:
θ=(XTX)−1XTy
4. Properties of the Normal
Equations
- Efficiency:
The normal equation provides a closed-form solution, which can be computed
in one step rather than iteratively.
- Computational Complexity:
The computation of (XTX)−1 can be computationally
expensive for large datasets, leading to potential numerical stability
issues.
5. Applications
The normal equations are used in:
- Linear Regression: To
find the optimal parameters.
- Machine Learning Models:
Many models leverage linear algebra formulations similar to the normal
equations.
6. Limitations
While the normal equations are
powerful, they have limitations:
- Inversion Problems: If
XTX is singular (non-invertible), it leads to
issues. This can occur when there is multicollinearity among features.
- Scalability:
For very large datasets, iterative approaches such as gradient descent may
be preferred due to computational constraints in computing the inverse.
Conclusion
The normal equations provide a
foundational method for performing linear regression, allowing practitioners to
derive model parameters efficiently when applicable conditions are met. More
intricate formulations and algorithms can build upon this foundation for
complex models and tasks in machine learning.
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