Struct TDistributionLoss 

pub struct TDistributionLoss { /* private fields */ }

Student’s t-distribution loss function (robust M-estimator).

The t-distribution loss is derived from the negative log-likelihood of Student’s t-distribution. It provides heavy tails for robustness against outliers, with the degrees of freedom parameter ν controlling the tail heaviness.

Mathematical Definition

ρ(s) = (ν + 1)/2 · log(1 + s/ν)
ρ'(s) = (ν + 1)/(2(ν + s))
ρ''(s) = -(ν + 1)/(2(ν + s)²)

where ν is the degrees of freedom and s = ||r||² is the squared residual norm.

Properties

• Heavy tails: Provides robustness through heavier tails than Gaussian
• Parameter control: Small ν → heavy tails (more robust), large ν → Gaussian (less robust)
• Well-founded: Based on maximum likelihood estimation with t-distribution
• Smooth: Continuous derivatives for all s > 0

Degrees of Freedom Selection

• ν = 3-4: Very robust, heavy outlier suppression
• ν = 5: Recommended default, good balance
• ν = 10: Moderate robustness
• ν → ∞: Converges to Gaussian (L2 loss)

Use Cases

• Robust regression with unknown outlier distribution
• SLAM and pose graph optimization with loop closure outliers
• Bundle adjustment with incorrect feature matches
• Any optimization problem with heavy-tailed noise

References

• Student’s t-distribution is widely used in robust statistics
• Applied in robust SLAM (e.g., Chebrolu et al. 2021, Agarwal et al.)

Example

use apex_solver::core::loss_functions::{LossFunction, TDistributionLoss};

let t_loss = TDistributionLoss::new(5.0)?;

let [rho, rho_prime, _] = t_loss.evaluate(4.0);
// Robust to outliers with heavy tails

Implementations

impl TDistributionLoss

pub fn new(nu: f64) -> ApexSolverResult<Self>

Create a new Student’s t-distribution loss function.

Arguments

• nu - Degrees of freedom (must be positive)

Recommended Values

• ν = 5.0: Default, good balance between robustness and efficiency
• ν = 3.0-4.0: More robust to outliers
• ν = 10.0: Less aggressive, closer to Gaussian

Trait Implementations

impl Clone for TDistributionLoss

fn clone(&self) -> TDistributionLoss

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for TDistributionLoss

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl LossFunction for TDistributionLoss

fn evaluate(&self, s: f64) -> [f64; 3]

Evaluate the loss function and its first two derivatives at squared residual s.