Non-uniform Linear Interpolation (NLI)

• NLI is a technique that approximates nonlinear functions via piecewise linear surrogates with adaptively placed breakpoints, optimizing error metrics like MSE and L1.
• Its algorithms, including dynamic programming and curvature-driven partitioning, enable efficient activation approximations that reduce computational error in neural networks.
• NLI offers practical hardware benefits by minimizing lookup table size, reducing latency and power consumption, and enhancing model explainability and performance.

Non-uniform Linear Interpolation (NLI) is a family of techniques for approximating nonlinear functions or integrals using piecewise linear surrogates with non-uniformly placed breakpoints and variable resolution across the input domain. NLI methods are distinguished from uniform linear interpolation by their data- or function-driven placement of segment boundaries, resulting in improved approximation properties and hardware efficiency. NLI provides core algorithmic and hardware advances for function approximation in high-performance machine learning inference, explainable AI, and scientific computing contexts.

1. Mathematical Formulations

Non-uniform linear interpolation proceeds by representing a continuous nonlinear function $f(x)$ on an interval $[a,b]$ via a set of $n+1$ adaptively chosen breakpoints $x_0<a<\dots<x_n=b$, yielding a piecewise linear approximation

$\hat f(x) = m_i x + b_i,\quad x \in [x_i, x_{i+1}]$

where $m_i = \frac{f(x_{i+1}) - f(x_i)}{x_{i+1} - x_i}$ and $b_i = f(x_i) - m_i x_i$ (Reggiani et al., 2023). For approximation outside $[x_0,x_n]$, endpoint segments may match the asymptotes of $f$.

Error metrics commonly optimized include:

• Mean squared error (MSE): $\frac{1}{b-a}\int_a^b (f(x) - \hat f(x))^2 dx$ (Reggiani et al., 2023)
• $L_1$-error: $\int_{x_{i-1}}^{x_i} |f(x) - l_i(x)| dx$, summed over segments (Gallego et al., 2013)
• Mean relative error over discrete sets, especially relevant for limited-precision hardware (Yu et al., 3 Feb 2026)

Optimal knot (breakpoint) placement is typically driven by function curvature. In the asymptotic, large-$N$ regime for smooth $f$, the optimal local knot density is proportional to $|f''(x)|^{1/3}$ for $L_1$ error (Gallego et al., 2013), while for MSE, heuristic and SGD-based approaches may be employed (Reggiani et al., 2023). Dynamic programming with the Bellman principle yields globally optimal cutpoint allocation under arbitrary error objectives (Yu et al., 3 Feb 2026).

2. Algorithms and Optimization Protocols

Multiple NLI algorithmic regimes exist depending upon the context:

• Dynamic Programming Global Search: Given a discrete grid $\{x_i\}$ and a fixed number $M+1$ cutpoints, NLI solves for cutpoint indices $\{i_k\}$ minimizing total (e.g., mean relative) error additively across $[x_{\min},x_{\max}]$. For error separability, Bellman's recurrence achieves the global solution in $O(MN^2)$ time (Yu et al., 3 Feb 2026).
• SGD and Heuristic Insert-Remove Procedures: For activation approximations (e.g., Flex-SFU), learnable knot parameters are updated using Adam to minimize MSE. A greedy remove-insert cycle eliminates least-useful breakpoints and refines segment boundaries, exploiting local error distributions (Reggiani et al., 2023).
• Curvature-driven Partitioning: For continuous $f$, practitioners compute a cumulative density function $$F(x) = \int_a^x |f''(t)|^{1/3} dt / \int_a^b |f''(t)|^{1/3} dt$$ and select breakpoints via inversion: $x_i = F^{-1}(i/N)$. This provides near-optimal error-equalized segments for $L_1$ minimization (Gallego et al., 2013).
• Integrated Gradient NLI: For explainable AI, non-uniformity is introduced by partitioning the interpolation path in parameter space into intervals reflecting local changes in prediction probability (Δ), with steps per interval $m_j \propto \sqrt{\Delta_j}$. Within each interval, subgrid steps are uniform (Bhat et al., 2023).

3. Application Domains

NLI is a foundational technique across a range of applications:

• Neural Network Nonlinear Layers: NLI efficiently replaces high-precision nonlinearities (e.g., SiLU, RMSNorm, Softmax exponentials, rsqrt) in LLM and DNN inference via dynamic-programming-optimized piecewise linear surrogates, enabling plug-in replacement with minimal accuracy drop (Yu et al., 3 Feb 2026, Reggiani et al., 2023).
• Model Explainability: For integrated gradients (IG), NLI dramatically reduces the convergence steps needed for faithful feature attributions by adaptively allocating integration resolution where the model output changes most, yielding $2.6{-}3.6\times$ runtime speedup for iso-convergence and negligible inference overhead (Bhat et al., 2023).
• Scientific and Numerical Computing: Optimally linearizing costly nonlinear operations, such as trigonometric or normalization functions, NLI provides error-predictable surrogates and real-time efficient evaluation, especially on GPUs (Gallego et al., 2013).

4. Hardware-Aware Implementation Strategies

NLI enables efficient hardware designs through:

• Segment Selection Structures: Binary-tree (log-depth) comparators decode the current interval index for non-uniform breakpoints, supporting scalable precision and high throughput (Reggiani et al., 2023).
• Fixed-latency Pipelining: A two-level address translation (macro/micro segmentation) minimizes critical path and comparator count, achieving single-cycle latency per activation and $\gtrsim 1$ Gops/s at 1 GHz (SMIC 28 nm) (Yu et al., 3 Feb 2026).
• Area, Power, and Throughput Efficiency: The segment partitioning reduces LUT size, area (down by 68–69% over uniform/NN-LUT baselines), and boosts energy efficiency 4-fold relative to state-of-the-art (Yu et al., 3 Feb 2026). Flex-SFU achieves throughputs of 1–4 acts/cycle for float and INT, with area overhead <6% in vector processors (Reggiani et al., 2023).

Table: Hardware Comparison for NLI Engine (Yu et al., 3 Feb 2026) | Method | LUT entries | Comparators | Multiplier | Adder | |----------|-------------|-------------|------------|-------| | NN-LUT | 256 | 256 | 1 | 1 | | NLI | 259 | 10 | 1 | 2 |

5. Theoretical and Empirical Error Analysis

NLI methods provide quantifiable approximation guarantees:

• The total $L_1$ error decays as $O(N^{-2})$ with optimal non-uniform placement, parameterized by the cubed integral of the local knot density (Gallego et al., 2013).
• NLI achieves 7–22.3× MSE reductions over uniform segmentation for activation functions (e.g., GELU, SiLU, tanh), and outperforms prior state-of-the-art schemes (Larkin’06, LowCost’20, Kim’22) for fixed segment budgets (Reggiani et al., 2023).

Sample results (Flex-SFU, 16 segments, sq-AAE reduction): | Func. | SoA error | Flex-SFU error | Improvement | |---------|-----------|---------------|-------------| | Tanh | $5.8{\times}10^{-6}$ | $4.3{\times}10^{-7}$ | 13.5× | | Sigmoid | $8.1{\times}10^{-7}$ | $1.2{\times}10^{-7}$ | 6.7× |

In the integrated gradients setting, NLI matches or betters vanilla IG on every convergence metric δ(m), requiring only $\approx$300–350 steps (vs. 800 for uniform) to reach δ_th=0.02, and up to 3.6× speedup for stricter thresholds (Bhat et al., 2023).

For large model inference, NLI yields near-zero accuracy drop compared to FP32 baselines, whereas quantization-insensitive NN-LUT approaches can degrade model accuracy and perplexity catastrophically (Yu et al., 3 Feb 2026).

6. Practical Guidelines and Limitations

• Interval Selection: Breakpoints in high curvature or information-dense regions ensure lower mean and max error (Bhat et al., 2023, Reggiani et al., 2023). For NLI in IG, $n_{int}=4\ldots8$ is empirically optimal.
• Optimizer Choice: SGD with Adam and greedy heuristics (remove-insert) are effective in practice for activation function surrogates (Reggiani et al., 2023).
• Integration Overhead: The pre-processing for breakpoint determination is negligible relative to total inference cost (≤3.2% in IG NLI (Bhat et al., 2023); setup amortized for hardware/firmware deployment).
• Scalability: DP-based methods scale quadratically in the number of input quantization points and linearly in segment count, limiting $M$ and grid $N$ for exhaustive search (Yu et al., 3 Feb 2026). Multi-level search or approximation may be required for ultra-high granularity.
• Deployment: Works on arbitrary differentiable models, all common floating/fixed-point formats, and is not data-dependent (calibration-free) (Yu et al., 3 Feb 2026).
• Extensions: Future directions include joint optimization with quantization schemes and adaptation to ultra-low-precision integer or BFLOAT16 deployment.

7. Comparative Impact and Significance

The adoption of non-uniform linear interpolation has empirically yielded:

• $2.6$–$3.6\times$ latency reduction at fixed attribution error in explainable AI (Bhat et al., 2023)
• 22.3× mean squared error reduction and 35.7% end-to-end DNN speedup for computer vision and NLP workloads (up to $3.3\times$ on specific models, Flex-SFU) (Reggiani et al., 2023)
• 4× improved energy efficiency and $>68\%$ area reduction for general nonlinear operator hardware (Yu et al., 3 Feb 2026)
• Statistically negligible (<0.01) accuracy loss across ImageNet, MMLU, GSM8k, HumanEval, and Wikitext-2 benchmarks when replacing analytic nonlinearities in modern LLMs (Yu et al., 3 Feb 2026)

Empirical evidence consistently demonstrates that non-uniform, function-adaptive cutpoint placement fundamentally outperforms uniform partitioning across accuracy, hardware utilization, and speed in nonlinear approximation tasks.

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Principal references: (Bhat et al., 2023, Reggiani et al., 2023, Yu et al., 3 Feb 2026, Gallego et al., 2013)