Inverse Distillation: Methods & Applications

• Inverse distillation is a meta-category that reverses traditional teacher-student roles or optimization objectives to improve model guidance.
• It employs diverse mechanisms such as adapter-tuning, inverse weighting, and inverse optimization to align teachers and students effectively.
• Applications range from image classification and diffusion models to LLM tuning and inverse problems, demonstrating versatile performance improvements.

Searching arXiv for papers on inverse distillation and closely related formulations. arxiv_search(query="inverse distillation OR inverse-distillation OR inverse knowledge distillation OR inverse bridge matching distillation", max_results=10, sort_by="submittedDate") Inverse distillation is a family of distillation procedures in which the inversion concerns the direction of transfer, the objective being optimized, the point in the pipeline where distillation occurs, or the relation between a forward model and an inverse task. In conventional knowledge distillation, a larger teacher transfers knowledge to a smaller student. Recent work uses the term for markedly different constructions: a smaller model teaching a larger one in adapter tuning (Ruan et al., 2024), an inverse min–max problem in diffusion, flow, bridge, and stochastic-interpolant compression (Kornilov et al., 26 Sep 2025), inference-time teacher-guided refinement during reverse sampling (Park et al., 2024), inverse-distilled discrete diffusion LLMs (Li et al., 22 Feb 2026), inverse mapping from logits to inputs in black-box distillation (Ma et al., 2022), inverse probability weighting for non-IID transfer sets (Niu et al., 2022), inverse difficulty temperature scaling in large-language-model KD (Xie et al., 13 Oct 2025), and maximization of clean–adversarial output divergence for transfer attacks (Wu et al., 24 Feb 2025). The term therefore denotes a recurring research pattern rather than a single standardized algorithm.

1. Semantic scope and recurring inversion patterns

Across the recent literature, “inverse distillation” has been used for several distinct but structurally related ideas. In some works, the inversion is a role reversal in teacher–student capacity; in others, it is an inverse optimization problem, a reversal of the usual agreement objective, or a distillation of forward-model guidance into inverse systems. This breadth is already visible from the domains in which the term appears: parameter-efficient fine-tuning, diffusion and bridge model acceleration, black-box classification KD, adversarial transfer, inverse imaging, inverse rendering, and protein design (Ruan et al., 2024, Kornilov et al., 26 Sep 2025, Ma et al., 2022, Wu et al., 24 Feb 2025, Melnyk et al., 2022).

Formulation	Inverse mechanism	Representative paper
Adapter-tuning KD	Smaller teacher, larger student	(Ruan et al., 2024)
Universal matching distillation	Inverse min–max over generator and fake model	(Kornilov et al., 26 Sep 2025)
Diffusion bridge compression	Inverse bridge matching from teacher drift to student coupling	(Gushchin et al., 3 Feb 2025)
Inference-time refinement	Teacher guidance applied after training during sampling	(Park et al., 2024)
Black-box KD	Learn an inverse-like mapping from logits to images	(Ma et al., 2022)
Transfer-gap correction	Inverse propensity weighting of KD loss	(Niu et al., 2022)
Token-adaptive LLM KD	Temperature decreases as difficulty increases	(Xie et al., 13 Oct 2025)
Adversarial transfer	Maximize, rather than minimize, soft-label discrepancy	(Wu et al., 24 Feb 2025)

A common thread is that the “inverse” move is not merely rhetorical. It changes the geometry of supervision. The reversal may be architectural, as in small-to-large teaching; variational, as in inverse optimization over induced distributions; procedural, as in teacher intervention at inference time; or task-structural, as in extracting fast, differentiable surrogates from forward predictors for inverse design problems. This suggests that inverse distillation is best understood as a meta-category of reversed supervision mechanisms.

2. Reversing supervision in discriminative distillation

The most literal reversal appears in inverse Distillation Adapter-Tuning (iDAT), where the smaller model is designated as the teacher and the larger model as the student (Ruan et al., 2024). The setting is adapter-tuning: the pre-trained backbone is frozen and trainable adapter modules acquire downstream task knowledge. For the Sequential Adapter,

$x^{\prime} = W_{up}(\sigma(W_{down}(x))) + x,$

and for the Parallel Adapter,

$x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$

The iDAT objective combines cross-entropy for teacher and student with logit-level MSE or KL distillation,

$$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$

$$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$

The paper argues that smaller and larger adapters acquire downstream knowledge with different statistical profiles: large ViT adapters are concentrated and sparse, whereas small ViT adapters are more dispersed and flatter. On a three-dataset study, ViT-S→ViT-B reached 83.37% mean accuracy, surpassing ViT-B→ViT-B at 82.11% and ViT-L→ViT-B at 83.03%. On VTAB-1K, the Sequential Adapter baseline achieved 71.55%, while iDAT-S-kl reached 74.21%, a 2.66% gain, with 0.20M trainable parameters instead of 0.13M (Ruan et al., 2024).

A second discriminative use of inversion appears in Inverse Probability Weighting Distillation, which does not reverse model size but reverses the weighting logic of KD under transfer-set bias (Niu et al., 2022). The paper distinguishes a human domain, providing hard targets, from a machine domain, providing teacher soft outputs, and argues that standard KD implicitly assumes IID transfer between them. The estimated propensity score is

$$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$

with inverse weight

$$\hat{w}(x) = \frac{1}{\hat{e}(x)} = 1 + \frac{H(kd, y)}{H(cls, y)}.$$

Only the KD term is reweighted:

$$L_{IPWD} = \sum_i \mathrm{CE}(y_i, p_s(x_i)) + \alpha\,T^2 \sum_i \hat{w}(x_i)\,\mathrm{KL}\big(p_t^T(x_i)\,\|\,p_s^T(x_i)\big).$$

On ImageNet, ResNet-50 → MobileNet-v1 improved from 70.49/89.92 with KD to 72.65/91.08 with IPWD; on CIFAR-100, ResNet50 → MobileNetV2 improved from 67.35 to 70.25 (Niu et al., 2022).

In LLM-oriented token-adaptive KD, the inverse move is temperature assignment. AdaKD defines a token difficulty score with the Hellinger distance and then applies Inverse Difficulty Temperature Scaling,

$$\tau_i = \tau_{base} \cdot \exp(-c \cdot \hat{s}_i.detach()).$$

Difficult tokens receive low temperatures for targeted error correction, while easy tokens receive high temperatures to encourage learning from the teacher’s full output distribution (Xie et al., 13 Oct 2025). Combined with Loss-Driven Adaptive Token Focusing, this produced consistent ROUGE-L gains: for Qwen2-7B → Qwen2-1.5B, RKD improved from 31.70 to 32.97; for OpenLLaMA2-7B → OpenLLaMA2-3B, GKD improved from 27.40 to 29.28; for GPT-2 1.5B → GPT-2 0.1B, GKD improved from 21.96 to 24.08 (Xie et al., 13 Oct 2025).

A different reversal under black-box constraints is Mapping-Emulation KD. Here the generator $$f_G: Y \simeq \mathbb{R}^C \to X \simeq \mathbb{R}^n$$ is trained as an inverse-like decoder of the teacher’s logits, and distillation aligns decoded images rather than logits directly (Ma et al., 2022). The step-2 objective is

$$L_{Dt} = \frac{1}{m} \sum_{i=1}^m \left\| G\!\left( \frac{S(x'^{(i)})}{\tau} \right) - G\!\left( \frac{T(x'^{(i)})}{\tau} \right) \right\|_F + \beta \frac{1}{m} \sum_{i=1}^m T(x'^{(i)}) \log\!\frac{T(x'^{(i)})}{S(x'^{(i)})}.$$

On CIFAR-100 with ResNet56→MobileNet, MEKD(soft) achieved 67.07% and MEKD(hard) 67.36%, versus DB3KD at 63.67%; on ImageNet-1K, RN50→RN34 reached 59.89% versus DB3KD at 58.61% (Ma et al., 2022).

In adversarial transfer, the reversal is objective-level. Inverse Knowledge Distillation augments the attack loss with a divergence term that maximizes disagreement between the surrogate’s outputs on benign and adversarial inputs,

$x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$0

with

$x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$1

This is “inverse” because conventional KD minimizes soft-label discrepancy, whereas IKD maximizes it (Wu et al., 24 Feb 2025). On ImageNet, RN50-based MIFGSM improved from 52.9 to 55.7 average black-box ASR, and VGG19BN-based DIFGSM improved from 51.0 to 57.8 (Wu et al., 24 Feb 2025).

3. Inverse optimization for matching, bridge, and one-step generators

A more abstract formulation appears in Universal Inverse Distillation and its real-data extension, RealUID (Kornilov et al., 26 Sep 2025). The central object is the Universal Matching loss,

$x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$2

which subsumes diffusion score matching, flow matching, bridge matching, and stochastic interpolants. Distillation is then framed as an inverse min–max:

$x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$3

RealUID augments this with real-data supervision through $x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$4, avoiding adversarial discriminators. The paper reports one-step generation at 18.636 ms per image on Ascend 910B3, compared with 30.745 ms for FGM/SiD, with 36.784M rather than 55.734M parameters. On CIFAR-10, unconditional RealUID achieved FID 2.03 and conditional RealUID achieved FID 1.91 (Kornilov et al., 26 Sep 2025).

Inverse Bridge Matching Distillation specializes the inverse-optimization view to Diffusion Bridge Models (Gushchin et al., 3 Feb 2025). Given a trained teacher DBM with drift $x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$5, the student generator induces a coupling $x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$6 and minimizes

$x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$7

The tractable surrogate is a difference of bridge-matching regression risks,

$x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$8

A notable feature is that training uses only corrupted inputs $x^{\prime} = s * W_{up}(\sigma(W_{down}(x))).$9, not paired clean targets. The method applies to both conditional and unconditional DBMs, supports one-step and multistep students, and reports 4×–100× speedups. On 4× super-resolution with I$$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$0SB teachers, IBMD 1-step achieved FID 2.5 versus teacher 2.8; on Edges→Handbags 64×64, IBMD at 2 NFE reached FID 0.67 (Gushchin et al., 3 Feb 2025).

Taken together, RealUID and IBMD establish a distinct sense of inverse distillation: the student is not merely trained to mimic a teacher trajectory. Instead, the generator is optimized so that the teacher remains optimal under the student-induced data or coupling distribution. This turns distillation into an inverse problem over induced path laws or induced matching objectives.

4. Inference-time and posterior-aware diffusion distillation

Distillation++ defines an inference-time rather than training-time form of inverse distillation (Park et al., 2024). A distilled diffusion student is refined during sampling by teacher-guided proximal updates based on Score Distillation Sampling. The central interpolation is

$$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$1

followed by a DDIM-like update. Distillation++ is data-free and post-training: it requires no additional source data and no fine-tuning. The paper applies only the first $$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$2 guided step in many settings and reports near-equal wall time between “4+1 steps” with guidance and “5 steps” without guidance: for LCM, 1.987s versus 1.996s. It improves FID, ImageReward, and PickScore across LCM, LCM-LoRA, SDXL-Lightning, SDXL-Lightning LoRA, DMD2, and SDXL-Turbo; for example, LCM improved from FID 20.674 to 20.149 and ImageReward 0.561 to 0.597 (Park et al., 2024).

In discrete diffusion LLMs, IDLM extends inverse distillation from continuous diffusion to the discrete setting and adds a uniqueness theorem (Li et al., 22 Feb 2026). The inverse objective is

$$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$3

and the paper proves that for SEDD, MDLM, and Duo,

$$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$4

with equality iff $$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$5. Practical training relies on simplex relaxation, Duo’s Gaussian reparameterization, and MDLM mask independence. Empirically, step counts are reduced from 1024→256 for SEDD, 1024→16 for MDLM, and 1024→16 for Duo, while preserving entropy and generative perplexity. Further distillation of Duo-DCD yields 1024→8 under Ancestral sampling and 1024→4 under Greedy-Tail, corresponding to 128× and 256× reductions relative to Duo (Li et al., 22 Feb 2026).

Noise Conditional Variational Score Distillation provides a posterior-centric form of inverse distillation for diffusion models (Peng et al., 11 Jun 2025). Its key identity introduces an effective noise level

$$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$6

and effective noisy input

$$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$7

so that the unconditional teacher score determines the denoising-posterior score. The student generative denoiser learns $$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$8 across many $$\mathcal{L}_{basic}=\mathcal{L}_{ce}(\hat{y_s},y)+\mathcal{L}_{ce}(\hat{y_t},y),$$9, enabling one-step generation, multistep refinement, and zero-shot inverse-problem inference via a Split Gibbs Sampler. On ImageNet-512×512, 4-step NCVSD-L achieved FID 1.76, surpassing EDM2-XXL at 1.81 and 2-step sCD-XXL at 1.88. On inverse problems, it reports LPIPS 0.128 for inpainting and 0.186 for phase retrieval with 50 NFEs (Peng et al., 11 Jun 2025).

These papers share a notable procedural inversion: distillation is no longer limited to offline compression of a teacher trajectory into a fixed student. It can be performed at inference time, or it can target denoising posteriors and reverse-time path measures rather than direct output matching.

5. Distilled guidance for inverse problems, rendering, and design

In inverse imaging, Deep Distillation Gradient Preconditioning learns a nonlinear preconditioner $$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$0 by matching the gradient geometry of a student inverse problem with ill-conditioned sensing matrix $$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$1 to that of a teacher inverse problem with better-conditioned synthetic matrix $$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$2 (Gualdrón-Hurtado et al., 6 Aug 2025). The preconditioned PnP-FISTA update is

$$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$3

followed by denoising or a proximal step. The distillation loss combines directional gradient matching, imitation, and supervision:

$$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$4

On single-pixel imaging, D$$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$5GP reached 34.55 PSNR versus 33.79 for supervised $$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$6 and 26.19 for a full-linear learned preconditioner; on MRI it achieved 29.39, and on super-resolution 25.88 (Gualdrón-Hurtado et al., 6 Aug 2025).

Repulsive Latent Score Distillation treats inverse problems through posterior sampling guided by pretrained diffusion priors (Zilberstein et al., 2024). The method interprets SDS as a Wasserstein gradient flow and augments it with repulsive multimodal variational approximations and an augmented latent–data posterior,

$$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$7

Repulsion mitigates mode collapse, and the augmented latent–data split addresses latent ambiguity. On half-face inpainting, RSD with $$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$8 obtained PSNR 24.69, LPIPS 0.111, FID 31.41, and Diversity 0.015, while RSD with $$\mathcal{L}_{all-kl}=\mathcal{L}_{basic}+\lambda_{kl}\mathcal{L}_{kl}(\hat{y_s},\hat{y_t}; \mathcal{T}).$$9 obtained 24.98, 0.109, 29.18, and 0.004; on random inpainting, RSD with $$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$0 reached PSNR 30.56, LPIPS 0.145, FID 41.11 (Zilberstein et al., 2024).

Progressive Radiance Distillation transfers knowledge in the opposite direction of conventional scene modeling: from an unstructured radiance field to physically interpretable material and lighting parameters (Ye et al., 2024). The blended rendering model is

$$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$1

where $$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$2 is a learned distillation progress map. Early in training $$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$3 is close to zero, preserving gradient sanity through the raw radiance field; as the physical model converges, $$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$4 increases, but it remains below one on pixels affected by unmodeled light paths. The method reports 39+8 minutes of training time and 221 FPS runtime on an RTX 4090, and the abstract states that it significantly outperforms state-of-the-art techniques quality-wise in both novel view synthesis and relighting (Ye et al., 2024).

In protein design, AlphaFold Distillation compresses AF2 confidence metrics into a fast, differentiable surrogate for inverse folding (Melnyk et al., 2022). A ProtBert-based student predicts pTM and pLDDT from sequence alone, and the inverse model is regularized with

$$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$5

The motivation is explicitly inverse: a forward folding model is too slow for inner-loop training of an inverse sequence model, so its confidence heads are distilled into a structure-consistency regularizer. The paper reports up to 3% improvement in sequence recovery and up to 45% improvement in protein diversity while maintaining structural consistency, and AFDistill inference is about 0.028 s for length 1024 and 0.035 s for length 2048 (Melnyk et al., 2022).

In these inverse-task settings, distillation is less about model compression in the narrow sense than about converting expensive or poorly structured guidance into an efficient optimization primitive. The student often becomes an auxiliary solver, preconditioner, renderer, or regularizer rather than a direct drop-in replacement for the teacher.

6. Limitations, sensitivities, and unresolved questions

A persistent source of confusion is terminological. “Inverse distillation” does not denote one canonical operation. It can mean small-to-large supervision in PEFT, inverse propensity weighting in transfer-gap correction, inverse difficulty temperature scaling in LLM KD, inverse mapping in black-box distillation, inverse matching in generative model compression, or maximization of discrepancy in adversarial attacks (Ruan et al., 2024, Niu et al., 2022, Xie et al., 13 Oct 2025, Ma et al., 2022, Kornilov et al., 26 Sep 2025, Wu et al., 24 Feb 2025). A common misconception is therefore to equate the term exclusively with reversed teacher size.

Theoretical guarantees are unevenly distributed. IDLM proves uniqueness of the minimizer under SEDD, MDLM, and Duo objectives, but the result assumes well-defined path measures and an optimal teacher (Li et al., 22 Feb 2026). RealUID gives a unified min–max theory and weighted-distance lemmas, but its empirical behavior depends sensitively on $$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$6, with the paper explicitly warning that configurations far from the recommended regime can degrade performance or destabilize training (Kornilov et al., 26 Sep 2025). D$$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$7GP provides empirical convergence analysis and a local Jacobian-based conditioning picture, while stating that formal convergence guarantees for PnP-FISTA with general nonlinear, iteration-dependent preconditioners remain open (Gualdrón-Hurtado et al., 6 Aug 2025).

Hyperparameter sensitivity is a recurring practical issue. In iDAT, KL temperature and $$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$8 matter, and appendix results favor $$\hat{e}(x) = \frac{H(cls, y)}{H(kd, y) + H(cls, y)},$$9 and $$\hat{w}(x) = \frac{1}{\hat{e}(x)} = 1 + \frac{H(kd, y)}{H(cls, y)}.$$0 over the default setting (Ruan et al., 2024). In AdaKD, the modulation strength $$\hat{w}(x) = \frac{1}{\hat{e}(x)} = 1 + \frac{H(kd, y)}{H(cls, y)}.$$1 is near-optimal on average, and LATF’s discrete ratio updates can cause small oscillations (Xie et al., 13 Oct 2025). In IKD, transferability deteriorates when $$\hat{w}(x) = \frac{1}{\hat{e}(x)} = 1 + \frac{H(kd, y)}{H(cls, y)}.$$2 becomes large, and the paper adopts $$\hat{w}(x) = \frac{1}{\hat{e}(x)} = 1 + \frac{H(kd, y)}{H(cls, y)}.$$3 as default (Wu et al., 24 Feb 2025). Distillation++ requires the student and teacher denoised estimates to live in the same latent space for direct interpolation, otherwise extra mapping steps are needed (Park et al., 2024).

Model and domain assumptions also remain important. iDAT is validated on image classification and does not present LLM experiments (Ruan et al., 2024). Progressive Radiance Distillation assumes static scenes, Cook–Torrance shading, environment-based lighting, and limited visibility modeling; indirect illumination and transparency remain difficult (Ye et al., 2024). RSD incurs $$\hat{w}(x) = \frac{1}{\hat{e}(x)} = 1 + \frac{H(kd, y)}{H(cls, y)}.$$4 pairwise cost for repulsion and is sensitive to kernel bandwidth (Zilberstein et al., 2024). AFDistill inherits AlphaFold2 biases and can degrade under distributional shift or skewed confidence labels (Melnyk et al., 2022).

A plausible implication is that future work will need a sharper taxonomy separating role-reversal distillation, inverse-objective distillation, inverse-optimization distillation, inference-time distillation, and inverse-task guidance distillation. The literature already shows that these categories can coexist, but it also shows that they rely on different assumptions, different mathematical objects, and different evaluation criteria.