This document explains how our library implements and extends Singular Learning Theory (SLT) in the context of quantum geometric learning.
In SLT, we consider statistical models as manifolds with singularities:
M = {p(x|w) | w ∈ W}
where:
- p(x|w) is a probability distribution
- w are the parameters
- W is the parameter space
Implementation:
statistical_manifold_t* create_statistical_manifold(
const manifold_params_t* params
) {
// Initialize manifold with singularities
statistical_manifold_t* M = init_singular_manifold();
// Add regular points
add_regular_points(M, params->regular_region);
// Add singular points
add_singular_points(M, params->singular_region);
return M;
}The RLCT characterizes the learning behavior near singularities:
λ = RLCT = min{β | ∫_W |L(w)|^{-β} dw < ∞}
Implementation:
double compute_rlct(
const loss_function_t* L,
const region_t* W,
const rlct_params_t* params
) {
// Initialize RLCT computation
rlct_computer_t* computer = init_rlct_computer(params);
// Compute integral for various β
for (double beta = params->min_beta;
beta <= params->max_beta;
beta += params->step) {
if (compute_integral(L, W, beta) < INFINITY) {
return beta;
}
}
return INFINITY;
}We extend SLT to quantum systems:
MQ = {ρ(w) | w ∈ W}
where ρ(w) are quantum density matrices.
Implementation:
quantum_statistical_manifold_t* create_quantum_manifold(
const quantum_params_t* params
) {
// Initialize quantum manifold
quantum_statistical_manifold_t* MQ =
init_quantum_manifold();
// Add quantum states
add_quantum_states(MQ, params->state_space);
// Add quantum singularities
add_quantum_singularities(MQ, params->singular_states);
return MQ;
}Extension of RLCT to quantum systems:
λQ = min{β | Tr[ρ(w)^{-β}] < ∞}
Implementation:
double compute_quantum_rlct(
const quantum_state_t* rho,
const parameter_space_t* W,
const quantum_rlct_params_t* params
) {
// Initialize quantum RLCT computation
quantum_rlct_computer_t* computer =
init_quantum_rlct_computer(params);
// Compute trace for various β
for (double beta = params->min_beta;
beta <= params->max_beta;
beta += params->step) {
if (compute_quantum_trace(rho, W, beta) < INFINITY) {
return beta;
}
}
return INFINITY;
}void singular_learning_step(
quantum_state_t* state,
const learning_params_t* params
) {
// Compute RLCT
double lambda = compute_quantum_rlct(state, params);
// Adjust learning rate based on RLCT
double adjusted_rate =
adjust_learning_rate(params->base_rate, lambda);
// Perform learning step
quantum_learning_step(state, adjusted_rate);
// Project back to manifold
project_to_statistical_manifold(state);
}void bayesian_singular_learning(
quantum_state_t* state,
const prior_t* prior,
const likelihood_t* likelihood,
const bayesian_params_t* params
) {
// Compute posterior near singularities
posterior_t* post = compute_singular_posterior(
state, prior, likelihood
);
// Perform Bayesian update
bayesian_update_near_singularity(state, post, params);
// Free resources
cleanup_posterior(post);
}model_t* select_model_with_singularities(
const model_set_t* models,
const data_t* data,
const selection_params_t* params
) {
// Initialize model selection
model_selector_t* selector =
init_singular_model_selector(params);
// Compute RLCT for each model
for (int i = 0; i < models->size; i++) {
double lambda = compute_model_rlct(
models->items[i], data
);
add_model_score(selector, lambda);
}
// Select optimal model
return select_optimal_model(selector);
}void analyze_learning_phases(
quantum_state_t* state,
const phase_params_t* params
) {
// Initialize phase analysis
phase_analyzer_t* analyzer =
init_phase_analyzer(params);
// Detect phase transitions
while (!analyzer->finished) {
// Compute RLCT
double lambda = compute_quantum_rlct(
state, analyzer->current_region
);
// Analyze phase
analyze_phase(analyzer, lambda);
// Move to next region
advance_region(analyzer);
}
}double numerical_rlct(
const loss_function_t* L,
const monte_carlo_params_t* params
) {
// Initialize Monte Carlo integration
mc_integrator_t* integrator =
init_monte_carlo(params);
// Compute RLCT through sampling
double rlct = compute_rlct_monte_carlo(
L, integrator
);
// Clean up
cleanup_monte_carlo(integrator);
return rlct;
}void analyze_singular_values(
const quantum_state_t* state,
const analysis_params_t* params
) {
// Compute singular values
singular_values_t* values =
compute_singular_values(state);
// Analyze distribution
analyze_value_distribution(values, params);
// Detect singularities
detect_singularities_from_values(values);
}-
Foundational SLT:
- Watanabe, S. (2009). "Algebraic Geometry and Statistical Learning Theory"
- Watanabe, S. (2001). "Algebraic Analysis for Nonidentifiable Learning Machines"
- Amari, S., Nagaoka, H. (2007). "Methods of Information Geometry"
-
Quantum Extensions:
- Nielsen, M.A., Chuang, I.L. (2010). "Quantum Computation and Quantum Information"
- Bengtsson, I., Życzkowski, K. (2017). "Geometry of Quantum States"
- Watanabe, S. (2010). "Asymptotic Learning Theory of Quantum Systems"
-
Geometric Methods:
- Arnold, V.I. (1989). "Mathematical Methods of Classical Mechanics"
- Mumford, D. (1976). "Algebraic Geometry I: Complex Projective Varieties"
- Griffiths, P., Harris, J. (1978). "Principles of Algebraic Geometry"
-
Statistical Learning:
- Vapnik, V.N. (1998). "Statistical Learning Theory"
- Amari, S. (2016). "Information Geometry and Its Applications"
- Bishop, C.M. (2006). "Pattern Recognition and Machine Learning"
-
Numerical Methods:
- Press, W.H., et al. (2007). "Numerical Recipes"
- Golub, G.H., Van Loan, C.F. (2013). "Matrix Computations"
- Robert, C.P., Casella, G. (2004). "Monte Carlo Statistical Methods"
-
Applications:
- Yamazaki, K., Watanabe, S. (2003). "Singularities in Mixture Models"
- Aoyagi, M., Watanabe, S. (2005). "Stochastic Complexities of Reduced Rank Regression"
- Drton, M., et al. (2009). "Lectures on Algebraic Statistics"
-
Recent Developments:
- Watanabe, S. (2018). "Mathematical Theory of Bayesian Statistics"
- Amari, S. (2020). "Information Geometry and Neural Networks"
- Various papers from Journal of Machine Learning Research and Annals of Statistics