Energy-Based Models

EBMs use the Helmholtz free energy as a discriminative score for novelty detection. For a given input x, the free energy E(x) is computed as the negative log of the partition function summed over all known classes. Lower energy indicates higher compatibility with the learned distribution. During inference, if E(x) exceeds a calibrated threshold, the sample is rejected as an unknown modulation scheme. This formulation naturally aligns with the conditional probability density learned by discriminative classifiers, allowing a standard neural network to be reinterpreted as an energy model without architectural changes.

Training EBMs involves contrastive methods that shape the energy landscape by contrasting positive examples (real training data) against negative examples (generated or noise samples). The objective is to push down energy on real modulation constellations while pushing up energy everywhere else. Key techniques include:

• Stochastic Gradient Langevin Dynamics (SGLD) for sampling negative examples from the model's current energy surface
• Noise contrastive estimation to avoid computing the intractable partition function
• Score matching which minimizes the gradient of the energy function rather than the energy itself This training paradigm explicitly creates an energy gap between known and unknown signal types.

A Joint Energy-Based Model unifies a discriminative classifier and a generative model within a single architecture. The same neural network simultaneously outputs class logits for known modulation types and defines an energy function over the input space. This dual nature provides:

• Improved calibration: The generative objective regularizes the classifier, producing better-calibrated probabilities
• Built-in OOD detection: The energy score serves directly as a novelty metric without additional heads or branches
• Hybrid discriminative-generative training that leverages both labeled and unlabeled data JEMs have demonstrated state-of-the-art performance on out-of-distribution detection benchmarks while maintaining competitive closed-set accuracy.

EBMs offer a principled alternative to SoftMax-based OOD detection. Unlike methods that rely on maximum SoftMax probability—which can produce overconfident predictions on unknown inputs—energy scores provide a density-aligned metric. The energy-based OOD score is computed as:

• E(x) = -T · log Σᵢ exp(fᵢ(x)/T) where fᵢ(x) are logits and T is temperature This score is theoretically connected to the log-likelihood of the input under the model's learned distribution. In spectrum monitoring applications, this means an EBM can reliably distinguish between a known QPSK signal and a novel modulation never seen during training, even when both produce similar SoftMax outputs.

Energy-based regularization explicitly shapes the energy surface during training to create a tight boundary around known classes. The training loss combines:

• Standard cross-entropy for correct classification of known modulation types
• Energy regularization term that penalizes low energy on out-of-distribution samples generated through SGLD or drawn from an auxiliary outlier dataset
• Margin-based ranking loss ensuring that in-distribution samples have energy below a margin m_in while OOD samples exceed a margin m_out This explicit energy shaping prevents the feature collapse problem where unknown signals inadvertently map to low-energy regions near known class prototypes.

EBMs enable iterative refinement of anomaly scores through Langevin dynamics sampling. Starting from an input signal's IQ samples, the model can perform gradient-based updates that move the sample toward lower-energy regions of the learned manifold. This process reveals:

• Reconstruction-based anomaly scores: The difference between the original and refined signal indicates the degree of novelty
• Energy trajectory analysis: The path taken during Langevin sampling provides a richer signal for OOD detection than a single energy evaluation
• Generative verification: Unknown modulation types can be visualized by observing how the model attempts to reconstruct them toward known constellations This capability is particularly valuable for spectrum forensics where analysts need to understand why a signal was flagged as anomalous.

The discriminative score derived from an energy-based model that separates in-distribution from out-of-distribution data. For a given input x, the free energy is computed as:

• Formula: E(x) = -T · log Σᵢ exp(-E(x, yᵢ)/T)
• Temperature parameter T controls the smoothness of the energy landscape
• Lower free energy indicates higher likelihood of belonging to known classes
• Unlike SoftMax probabilities, free energy does not require normalization over all possible classes, making it inherently suitable for open set rejection

An alternative training objective for energy-based models that avoids the intractable partition function by matching the gradient of the log-density (score) rather than the density itself. Key properties:

• Minimizes the Fisher divergence between model and data score functions
• Denoising score matching adds noise to data points and learns to denoise, providing a tractable objective
• Forms the theoretical foundation for diffusion models and score-based generative modeling
• Enables energy function learning without requiring negative sampling or MCMC

A unified framework where a single neural network outputs an energy value for any (input, label) pair, enabling simultaneous classification and generation. For open set recognition:

• Discriminative mode: Predict class by finding the label that minimizes energy
• Generative mode: Sample new inputs via Langevin dynamics on the energy landscape
• The marginal energy E(x) = -log Σᵧ exp(-E(x, y)) serves as a novelty score
• Outperforms hybrid discriminative-generative models by maintaining a calibrated energy surface across all inputs

A Markov Chain Monte Carlo method used to sample from the probability distribution defined by an energy-based model. The iterative update rule:

• xₜ₊₁ = xₜ - ε · ∇ₓE(xₜ) + √(2ε) · N(0, I)
• Combines gradient descent on the energy with injected Gaussian noise
• The noise term prevents collapse to local minima and ensures proper exploration
• Used at inference time to generate synthetic signal samples or refine noisy IQ inputs toward low-energy configurations

A training technique that transforms density estimation into a binary classification problem between real data and a known noise distribution. For energy-based models:

• The model learns to distinguish true signal samples from artificially generated noise
• Avoids computing the partition function by treating it as a learnable parameter
• The ratio of model density to noise density yields the unnormalized energy
• Particularly effective when the noise distribution is chosen to match the support of the data manifold