Bum Jun Kim
Publications
SEAM: Global consistency beyond local accuracy in scientific machine learning
Scientific machine learning commonly validates models at the level of a subdomain, a benchmark split, or an explanation for one prediction. Yet such local checks cannot establish whether the resulting explanations can be assembled into one globally admissible explanation. We introduce Scientific Explanation-Admissibility Machines (SEAM), a generator-agnostic framework that makes this local-to-global consistency question computable across regions, sensors, regimes, and model components. The finite explanation-sheaf instantiation SEAM-$Ω$ represents each region by a structured explanation with state, closure, and observation channels together with optional contract metadata; compares neighboring explanations on their overlaps; and converts disagreement into a channel-resolved obstruction. This obstruction locates inconsistency and tests competing declared accounts by restricting each repair to the revisions that one account permits. Exact feasibility refutes or retains an account; when exact repair is unavailable, residual-aware regularized records provide a separately labeled empirical attribution. The framework also separates inconsistency from non-identifiability and monitors learned generators under distribution shift. We establish theorems for minimum-cost intervention and conservation-contract detectability, together with companion results for identifiability and closure recoverability. Across nineteen experiments involving synthetic partial differential equation systems and out-of-distribution Fourier neural operator (FNO) monitoring, SEAM detects incompatible explanations even when local predictions are accurate, and attributes failures to specific channels and overlaps. SEAM adds a global explanation-consistency audit to existing solvers and learning models, testing whether their local explanations form a coherent scientific account.
Looped Transformers with Source-Centered State Evolution
Looped Transformers create a useful train- and test-time compute axis by reusing the same Transformer block over recurrent depth, increasing effective depth at a fixed parameter count. However, that shared block must then govern an entire trajectory of varying hidden states over trained and extrapolated depths. Furthermore, in additive-injection looped Transformers, an input-conditioned signal is reintroduced at every recurrent step, so applying the shared transition at an input-conditioned reference can still move the hidden state. In this paper, we propose Source-Centered State Evolution (SCSE), which is designed to reconcile input conditioning with reference-preserving shared recurrence. Specifically, SCSE retains input dependence through its learned anchor and initial deviation, allows nonzero deviations to drive recurrent computation while mapping zero deviation to zero, and guarantees exact anchor invariance through its zero-deviation mask. The designated anchor is thereby a one-step fixed point by construction. The zero-deviation forcing bias is the next deviation produced from the anchor itself and vanishes in SCSE, while nonzero deviations remain active and support state-dependent recurrent computation. Our theory shows that the zero-deviation forcing bias is a design degree of freedom whose task effect can be harmful, neutral, or beneficial; SCSE resolves this choice in favor of exact anchor invariance by setting the bias to zero. Across WikiText-2, WikiText-103, direct web-corpus pretraining, held-out web-text transfer, and LAMBADA completion, SCSE improves the controlled recurrent quality frontier. Ablation studies identify the learned anchor and the anchor-coordinate deviation recurrence as the primary contributors to the gain, and a trained-model case study grounds the anchor-response diagnostic in observed recurrent motion.
Lottery Tickets Are Not Deployment Tickets
Reports on how sparsification, compression, and lottery tickets change model behavior have been mixed in the prior literature, with beneficial effects observed in some studies and adverse effects in others. Moreover, prior work has not considered actual deployment conditions, where decision logic is already fixed for the incumbent. To assess these mixed findings from a practical standpoint, we study the production-replacement question at the deployment level, namely whether an accuracy-matched lottery ticket or another sparse challenger can replace an incumbent dense model without reconfiguring downstream decision logic. We therefore audit a broad, protocol-specific panel of deployment-relevant behaviors spanning calibration, OOD response, class-level reliability, representations, and downstream policy decisions, and summarize clean-accuracy-excluded deviations with a behavioral-compatibility distance. Across extensive experiments, sparse candidates repeatedly recover dense-reference accuracy yet remain behaviorally different; in several study-band-matched settings, LTs also show lower corruption accuracy. In small-gap settings with fixed-threshold policy diagnostics, lottery-ticket replacement changes 7% to 10% of accept--review decisions. This churn creates precisely the burden that drop-in replacement is meant to avoid: reconfiguring and revalidating downstream decision logic. These findings establish the limits of clean-accuracy certification: Establishing compatibility with a fixed incumbent is distinct from attributing churn uniquely to sparsity or treating every measured deviation as harmful. Our theory explains the routing result: Even exact pointwise top-1 agreement cannot bound fixed-threshold decision changes, and small confidence shifts near the operating boundary can generate first-order routing churn.
Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation
Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes. While powerful, this perspective is incomplete: it primarily captures complexity through regularity, and therefore does not distinguish intuitively simple and complicated objects with comparable regularity, such as the square-root function and a typical Brownian path. The guiding message is that neural networks should be viewed not only as flexible basis functions, but also as models of computation. If a function is computable by a real-valued circuit over a prescribed elementary gate language, then it can be computed to comparable accuracy by an NN with explicit depth, width, and non-zero-parameter bounds controlled by the depth, width, gate count, and gate structure. Thus, neural-network complexity is not governed by regularity alone, but also by algorithmic complexity. We then show that any definable NN model satisfying a natural parallelization condition, allowing possibly multivariate non-linearities such as attention or layer normalization, is a universal approximator if and only if it contains a non-affine nonlinearity. The scope of our theory is illustrated by deducing universal approximation guarantees for continuous functions, minimax-optimal approximation guarantees for Besov classes, logarithmic-error complexity for holomorphic functions, and by showing that NNs can emulate numerical algorithms such as Newton-Raphson root finding and power iteration without architecture-specific arguments. Its precision is illustrated by shortest-path computation on $k$-vertex graphs: compiling the tropical dynamic-programming circuit yields NNs with O(log(1/ε)) non-zero parameters, exponentially improving in 1/ε over the generic $O(ε^{-c k^2})$ Lipschitz-approximation scale, for a constant c>0.
Residual Koopman Spectral Profiling for Predicting and Preventing Transformer Training Instability
Training divergence in transformers wastes compute, yet practitioners discover instability only after expensive runs begin. They therefore need an expected probability of failure for a transformer before training starts. Our study of Residual Koopman Spectral Profiling (RKSP) provides such an estimate. From a single forward pass at initialization, RKSP extracts Koopman spectral features by applying whitened dynamic mode decomposition to layer-wise residual snapshots. Our central diagnostic, the near-unit spectral mass, quantifies the fraction of modes concentrated near the unit circle, which captures instability risk. For predicting divergence across extensive configurations, this estimator achieves an AUROC of 0.995, outperforming the best gradient baseline. We further make this diagnostic actionable through Koopman Spectral Shaping (KSS), which reshapes spectra during training. We empirically validate that our method works in practice: RKSP predicts divergence at initialization, and when RKSP flags high risk, turning on KSS successfully prevents divergence. In the challenging high learning rate regime without normalization layers, KSS reduces the divergence rate from 66.7% to 12.5% and enables learning rates that are 50% to 150% higher. These findings generalize to WikiText-103 language modeling, vision transformers on CIFAR-10, and pretrained language models, including GPT-2 and LLaMA-2 up to 7B, as well as emerging architectures such as MoE, Mamba-style SSMs, and KAN.