Formula Pattern Catalog

Use this catalog when a paper equation is important but the notation is unfamiliar. The goal is to rewrite the equation into a known pattern, define the sampled unit, and connect it to the claim.

Most paper formulas can be reduced to:

$$\text{quantity}={\mathrm{aggregate}}_{u\sim q(u)}\left[{\text{operation}}_{\theta }(u)\right]$$

where $u$ is the sampled unit, $q(u)$ is the sampling distribution, and ${\text{operation}}_{\theta }$ is a model, loss, score, update, estimator, or metric.

Pattern Map

Pattern	Canonical Form	Main Question
Empirical risk	$\frac{1}{n}{\sum }_i\mathcal{L}(f_{\theta }(x_i),y_i)$	what unit, label, and loss are optimized?
Maximum likelihood	${\sum }_i\log p_{\theta }(x_i)$	what distribution is modeled?
Conditional likelihood	${\sum }_i\log p_{\theta }(y_i\mid x_i)$	what is conditioned on and what is predicted?
Cross entropy	$-{\sum }_k{y}_k\log p_{\theta ,k}$	are targets hard labels, soft labels, or distributions?
Contrastive loss	$-\log \frac{\exp (s(a,p)/\tau )}{{\sum }_j\exp (s(a,j)/\tau )}$	what are anchor, positive, negative set, and temperature?
Reconstruction	$\Vert x-{\hat{x}}_{\theta }\Vert$	what information is allowed through the bottleneck?
ELBO	${\mathbb{E}}_{q_{\varphi }(z\mid x)}[\log p_{\theta }(x\mid z)]-\mathrm{KL}(q_{\varphi }(z\mid x)\Vert p(z))$	what is latent, decoder, prior, and approximation?
Denoising	$\Vert ϵ-{ϵ}_{\theta }(x_t,t,c){\Vert }^2$	what noise level and target are used?
Flow matching	$\Vert v_{\theta }(x_t,t,c)-u_t{\Vert }^2$	what path and velocity target define training?
Score matching	$\Vert s_{\theta }(x_t,t)-{\nabla }_x\log p_t(x_t){\Vert }^2$	what score is estimated and how is it sampled?
Message passing	$h_i^{\prime }=\varphi (h_i,{\square }_{j\in \mathcal{N}(i)}\psi (h_i,h_j,e_{ij}))$	what graph, neighbors, messages, and aggregation are used?
Attention	$\mathrm{softmax}(Q{K}^{\top }/\sqrt{d_k}+M)V$	what tokens attend to what, under which mask?
Equivariance	$\varphi (g\cdot x)=\rho (g)\varphi (x)$	what group acts and how should output transform?
Metric estimate	$\frac{1}{m}{\sum }_{j}M({\hat{y}}_j,y_j)$	what examples, selection rule, and denominator define the score?

Supervised Objective

The default supervised pattern is:

$$\hat{\theta }=\arg \underset{\theta }{\min }\frac{1}{n}\sum _{i=1}^n\mathcal{L}\left(f_{\theta }(x_i),y_i\right)$$

• $x_i$: input object or representation.
• $y_i$: label, target, class, scalar, ranking, coordinates, or structured output.
• $f_{\theta }$: model.
• $\mathcal{L}$: surrogate loss.
• $n$: training examples after sampling and filtering.

Checks:

• Does $x_i$ mean raw input or processed representation?
• Does $y_i$ include context such as target, assay, time, or condition?
• Is $\mathcal{L}$ the same object as the final metric?
• Is the final test set untouched by model selection?

Likelihood and Conditional Likelihood

Unconditional likelihood:

$$\hat{\theta }=\arg \underset{\theta }{\max }\sum _{i=1}^n\log p_{\theta }(x_i)$$

Conditional likelihood:

$$\hat{\theta }=\arg \underset{\theta }{\max }\sum _{i=1}^n\log p_{\theta }(y_i\mid x_i,c_i)$$

• $x_i$: observed object or condition.
• $y_i$: predicted object.
• $c_i$: optional context such as prompt, target, pocket, assay, or class.

Checks:

• Is the paper modeling data likelihood, label likelihood, or sequence likelihood?
• Is likelihood tractable or replaced by a surrogate bound?
• Does high likelihood imply the reported utility claim?

Contrastive Objective

A common contrastive form is:

$${\mathcal{L}}_i=-\log \frac{\exp (\mathrm{sim}(z_i,z_i^{+})/\tau )}{\exp (\mathrm{sim}(z_i,z_i^{+})/\tau )+{\sum }_{j\in {\mathcal{N}}_i}\exp (\mathrm{sim}(z_i,z_j^{-})/\tau )}$$

• $z_i$: anchor representation.
• $z_i^{+}$: positive representation.
• $z_j^{-}$: negative representation.
• ${\mathcal{N}}_i$: negative set.
• $\tau$: temperature.

Checks:

• What makes a positive pair valid?
• Are negatives true negatives, in-batch negatives, hard negatives, or false negatives?
• Does the downstream evaluator test representation quality or only the contrastive setup?

Generative Path Objective

Diffusion, flow matching, and score models usually define a path:

$$x_0\sim p_{\mathrm{data}},x_t\sim p_t(x_t\mid x_0),\mathcal{L}(\theta )={\mathbb{E}}_{x_0,t,x_t}\left[\Vert g_{\theta }(x_t,t,c)-g^{\star }(x_t,t,x_0,c){\Vert }^2\right]$$

• $x_0$: clean data sample.
• $x_t$: noised or interpolated state.
• $t$: time or noise level.
• $c$: condition.
• $g_{\theta }$: predicted noise, score, velocity, or clean sample.
• $g^{\star }$: training target.

Checks:

• Is the target noise, score, velocity, $x_0$, or another parameterization?
• What sampling solver turns the learned field into a sample?
• Are validity, novelty, diversity, and utility evaluated separately from the training loss?

Graph and Set Aggregation

Graph equations often hide permutation assumptions:

$$m_i={\square }_{j\in \mathcal{N}(i)}{\psi }_{\theta }(h_i,h_j,e_{ij}),h_i^{\prime }={\varphi }_{\theta }(h_i,m_i)$$

• $\mathcal{N}(i)$: neighbors of node $i$.
• ${\psi }_{\theta }$: message function.
• $\square$: permutation-invariant aggregation such as sum, mean, max, or attention.
• ${\varphi }_{\theta }$: update function.

Checks:

• What is a node: atom, residue, token, region, candidate, or object?
• What is an edge: bond, contact, distance threshold, sequence adjacency, or learned relation?
• Does the aggregation preserve the required permutation behavior?

Coordinate and Symmetry Rule

Coordinate-heavy models should state the transformation rule:

$$x_i^{\prime }=R{x}_i+t,d_{ij}=\Vert x_i-x_j{\Vert }_2$$

For equivariant outputs:

$$\varphi (RX+t)=R\varphi (X)$$

For invariant scalar outputs:

$$f(RX+t)=f(X)$$

Checks:

• Is the target scalar, vector, coordinate, force, velocity, or field?
• Is the group translation, rotation, reflection, permutation, $SO(3)$, $SE(3)$, or $E(3)$?
• Does preprocessing leak a coordinate frame unavailable at inference?

Metric and Estimator

Reported scores are estimators:

$$\hat{M}=\frac{1}{m}\sum _{j=1}^{m}M({\hat{y}}_j,y_j)$$

• $m$: evaluated examples after filtering.
• $M$: metric function.
• ${\hat{y}}_j$: prediction after thresholding, ranking, sampling, or selection.
• $y_j$: reference label or object.

Checks:

• Is the metric averaged per example, per target, per scaffold, per dataset, or per task?
• Was the threshold, checkpoint, or sampler selected on validation data?
• Are confidence intervals, paired comparisons, or seed variation needed?

Claim Routing

Formula Looks Like	Route First
supervised loss	Machine learning, Loss function
likelihood or KL	Information and likelihood
contrastive or masked objective	Learning methods
diffusion, score, or flow objective	Generative models
graph message passing	Architectures, Discrete math and graphs
coordinate transform	Geometry and symmetry
reported metric	Evaluation math
mixed AI and domain claim	AI Computational Biology Math contract

Related

• Formula intake
• Formula explanation ladder
• Information and likelihood
• Dynamical systems
• Evaluation math
• AI paper intake
• AI Computational Biology Math contract