A Higher-Order Ising Model with Gradient-Free Update.

Area of Science:

• Statistical mechanics
• Machine learning
• Complex systems

Background:

• The conventional Ising model can store patterns as stable states but has limitations.
• Its energy function is restricted to second-order polynomials, limiting its capacity for complex pattern storage.
• A significant gap exists in constructing Ising models where desired patterns are stable solutions.

Purpose of the Study:

• To extend the conventional Ising model for broader applications.
• To overcome the limitations of second-order polynomial energy functions in Ising models.
• To develop a model that can store complex patterns beyond the Hebbian rule's capabilities.

Main Methods:

• The extended Ising model is defined by combining desired patterns in a product.
• The model's energy function has no limit on its order.
• Explicit closed-form update formulas are derived, avoiding gradient evaluation.

Main Results:

• The extended Ising model demonstrates enhanced pattern storage capabilities.
• The derived update formulas allow for direct computation without network training.
• The update algorithm converges to a local minimum in finite steps.

Conclusions:

• The extended Ising model offers a powerful alternative to conventional Ising networks.
• It enables the storage of complex patterns previously unachievable.
• The model's efficiency and lack of need for training open new avenues for research and application.