We develop a unified mathematical framework for certified Top-$k$ attention truncation that quantifies approximation error at both the distribution and output levels. For a single attention distribution $P$ and its Top-$k$ truncation $\hat P$, we show that the total-variation distance coincides with the discarded softmax tail mass and satisfies $\mathrm{TV}(P,\hat P)=1-e^{-\mathrm{KL}(\hat P\Vert P)}$, yielding sharp Top-$k$-specific bounds in place of generic inequalities. From this we derive non-asymptotic deterministic bounds -- from a single boundary gap through multi-gap and blockwise variants -- that control $\mathrm{TV}(P,\hat P)$ using only the ordered logits. Using an exact head-tail decomposition, we prove that the output error factorizes as $\|\mathrm{Attn}(q,K,V)-\mathrm{Attn}_k(q,K,V)\|_2=τ\|μ_{\mathrm{tail}}-μ_{\mathrm{head}}\|_2$ with $τ=\mathrm{TV}(P,\hat P)$, yielding a new head-tail diameter bound $\|\mathrm{Attn}(q,K,V)-\mathrm{Attn}_k(q,K,V)\|_2\leτ\,\mathrm{diam}_{H,T}$ and refinements linking the error to $\mathrm{Var}_P(V)$. Under an i.i.d. Gaussian score model $s_i\sim\mathcal N(μ,σ^2)$ we derive closed-form tail masses and an asymptotic rule for the minimal $k_\varepsilon$ ensuring $\mathrm{TV}(P,\hat P)\le\varepsilon$, namely $k_\varepsilon/n\approxΦ_c(σ+Φ^{-1}(\varepsilon))$. Experiments on bert-base-uncased and synthetic logits confirm the predicted scaling of $k_\varepsilon/n$ and show that certified Top-$k$ can reduce scored keys by 2-4$\times$ on average while meeting the prescribed total-variation budget.

We summarize the data statistics in our experiments in Table 1. For both fully and semi-supervised node classification tasks on the citation networks, Cora, Citeseer and Pubmed, we train our DGC following the hyper-parameters in SGC [5]. Specifically, we train DGC for 100 epochs using Adam [2] with learning rate 0.2. For weight decay, as in SGC, we tune this hyperparameter on each dataset using hyperopt [1] for 10,000 trails. For the large-scale inductive learning task on the Reddit network, we also follow the protocols of SGC [5], where we use L-BFGS [3] optimizer for 2 epochs with no weight decay.

We use the PyTorch framework for our experiments. Similar to TD3, we implement our GRU-ODE in SAC. In this ablation study, we ask two questions in relation to numerical integration. Thus, simple numerical solvers are enough. We evaluate the time costs of different baselines on Walker-P environments.

The main contribution of the paper is that it shows that Runge-Kutta solutions of order 1, 2, 3 can be obtained from integrated Weiner processes.

We summarize the data statistics in our experiments in Table 1. For both fully and semi-supervised node classification tasks on the citation networks, Cora, Citeseer and Pubmed, we train our DGC following the hyper-parameters in SGC [5]. Specifically, we train DGC for 100 epochs using Adam [2] with learning rate 0.2. For weight decay, as in SGC, we tune this hyperparameter on each dataset using hyperopt [1] for 10,000 trails. For the large-scale inductive learning task on the Reddit network, we also follow the protocols of SGC [5], where we use L-BFGS [3] optimizer for 2 epochs with no weight decay.

Image samples are shown in Section I.

Despite the recent proliferation of machine learning methods like SINDy that promise automatic discovery of governing equations from time-series data, there remain significant challenges to discovering models from noisy datasets. One reason is that the linear regression underlying these methods assumes that all noise resides in the training target (the regressand), which is the time derivative, whereas the measurement noise is in the states (the regressors). Recent methods like modified-SINDy and DySMHO address this error-in-variable problem by leveraging information from the model's temporal evolution, but they are also imposing the equation as a hard constraint, which effectively assumes no error in the regressand. Without relaxation, this hard constraint prevents assimilation of data longer than Lyapunov time. Instead, the fulfilment of the model equation should be treated as a soft constraint to account for the small yet critical error introduced by numerical truncation. The uncertainties in both the regressor and the regressand invite the use of orthogonal distance regression (ODR). By incorporating ODR with the Bayesian framework for model selection, we introduce a novel method for model discovery, termed ODR-BINDy, and assess its performance against current SINDy variants using the Lorenz63, Rossler, and Van Der Pol systems as case studies. Our findings indicate that ODR-BINDy consistently outperforms all existing methods in recovering the correct model from sparse and noisy datasets. For instance, our ODR-BINDy method reliably recovers the Lorenz63 equation from data with noise contamination levels of up to 30%.