arXiv subjects

Jeffrey Uhlmann

Publications and source records attributed to Jeffrey Uhlmann.

At least 19 recordsLinked to original sources

arXiv2026

A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general solution to a longstanding open problem relevant to a wide variety of applications in robotics, tracking, and control systems. The new inverse complements the Drazin inverse (which is consistent with respect to similarity transformations) and the Moore-Penrose inverse (which is consistent with respect to unitary/orthonormal transformations) to complete a trilogy of generalized matrix inverses that exhausts the standard family of analytically-important linear system transformations. Results are generalized to obtain unit-consistent and unit-invariant matrix decompositions and examples of their use are described.

math.NA↗

arXiv2026

Unit-Consistent (UC) Adjoint for GSD and Backprop in Deep Learning Applications

Deep neural networks constructed from linear maps and positively homogeneous nonlinearities (e.g., ReLU) possess a fundamental gauge symmetry: the network function is invariant to node-wise diagonal rescalings. However, standard gradient descent is not equivariant to this symmetry, causing optimization trajectories to depend heavily on arbitrary parameterizations. Prior work has proposed rescaling-invariant optimization schemes for positively homogeneous networks (e.g., path-based or path-space updates). Our contribution is complementary: we formulate the invariance requirement at the level of the backward adjoint/optimization geometry, which provides a simple, operator-level recipe that can be applied uniformly across network components and optimizer state. By replacing the Euclidean transpose with a Unit-Consistent (UC) adjoint, we derive UC gauge-consistent steepest descent and backprogation.

cs.LG↗

arXiv2025

Graded Projection Recursion (GPR): Corrections, Obstructions, and Conservative Approximate Matrix Multiplication

Earlier versions proposed Graded Projection Recursion (GPR) as a deterministic packed-recursion framework for model-honest near-quadratic dense matrix multiplication. This revised version withdraws the exact dense matrix multiplication theorem and the downstream consequences that depended on it with a conservative AMM framework. The local ingredients remain useful as local tools: the three-band packing identity, scaled middle-band extraction under certified gaps, centering and reconstruction identities, and row/column normalization bounds. The gap in the earlier argument is global: the proof relied on a bounded active-state realization that would remove first-mismatch terms through the recursion. For arbitrary dense inputs this would require an exact equality filter over the inner index. We formulate this obstruction as a target-slice/equality-filter problem and give a rank/capacity argument against the natural separable active-state realization. The positive replacement is a conservative approximate matrix multiplication framework. For chosen protected left and right query subspaces, the low/marginal part of AB is computed exactly and an unbiased AMM primitive is applied only to the high/high residual. The resulting estimator is unbiased, preserves protected queries exactly in every realization, localizes stochastic error to the residual subspace, and inherits residual output-norm or query-risk guarantees from the underlying estimator.

cs.CC↗

arXiv2025

The Fourth State: Signed-Zero Ternary for Stable LLM Quantization (and More)

Quantization is usually regarded as a means to trade quality of performance for reduced compute requirements, i.e., as a suboptimal approximation. However, if examined in terms of a fixed overall resource budget, a very different perspective arises. We introduce Signed-Zero Ternary (SZT), a 2-bit quantization that deterministically provides gradient information with no forward-path penalty. Our analysis provides evidence that it may improve information density compared to non-quantized alternatives.

cs.LG↗

arXiv2024

Optimal Structured Matrix Approximation for Robustness to Incomplete Biosequence Data

We propose a general method for optimally approximating an arbitrary matrix $\mathbf{M}$ by a structured matrix $\mathbf{T}$ (circulant, Toeplitz/Hankel, etc.) and examine its use for estimating the spectra of genomic linkage disequilibrium matrices. This application is prototypical of a variety of genomic and proteomic problems that demand robustness to incomplete biosequence information. We perform a simulation study and corroborative test of our method using real genomic data from the Mouse Genome Database. The results confirm the predicted utility of the method and provide strong evidence of its potential value to a wide range of bioinformatics applications. Our optimal general matrix approximation method is expected to be of independent interest to an even broader range of applications in applied mathematics and engineering.

stat.AP↗

arXiv2024

Rank-Preference Consistency as the Appropriate Metric for Recommender Systems

In this paper we argue that conventional unitary-invariant measures of recommender system (RS) performance based on measuring differences between predicted ratings and actual user ratings fail to assess fundamental RS properties. More specifically, posing the optimization problem as one of predicting exact user ratings provides only an indirect suboptimal approximation for what RS applications typically need, which is an ability to accurately predict user preferences. We argue that scalar measures such as RMSE and MAE with respect to differences between actual and predicted ratings are only proxies for measuring RS ability to accurately estimate user preferences. We propose what we consider to be a measure that is more fundamentally appropriate for assessing RS performance, rank-preference consistency, which simply counts the number of prediction pairs that are inconsistent with the user's expressed product preferences. For example, if an RS predicts the user will prefer product A over product B, but the user's withheld ratings indicate s/he prefers product B over A, then rank-preference consistency has been violated. Our test results conclusively demonstrate that methods tailored to optimize arbitrary measures such as RMSE are not generally effective at accurately predicting user preferences. Thus, we conclude that conventional methods used for assessing RS performance are arbitrary and misleading.

cs.IR↗

arXiv2024

Gaussianity and the Kalman Filter: A Simple Yet Complicated Relationship

One of the most common misconceptions made about the Kalman filter when applied to linear systems is that it requires an assumption that all error and noise processes are Gaussian. This misconception has frequently led to the Kalman filter being dismissed in favor of complicated and/or purely heuristic approaches that are supposedly "more general" in that they can be applied to problems involving non-Gaussian noise. The fact is that the Kalman filter provides rigorous and optimal performance guarantees that do not rely on any distribution assumptions beyond mean and error covariance information. These guarantees even apply to use of the Kalman update formula when applied with nonlinear models, as long as its other required assumptions are satisfied. Here we discuss misconceptions about its generality that are often found and reinforced in the literature, especially outside the traditional fields of estimation and control.

math.OC↗

arXiv2024

Efficient Search in Graph Edit Distance: Metric Search Trees vs. Brute Force Verification

This report evaluates the efficiency of Graph Edit Distance (GED) computation for graph similarity search, comparing Cascading Metric Trees (CMT) with brute-force verification. Despite the anticipated advantages of CMT, our findings indicate it does not consistently outperform brute-force methods in speed. The study, based on graph data from PubChem, suggests that the computational complexity of GED-based GSS remains a challenge.

cs.DB↗

arXiv2023

A Cautionary Example Relating to the Interpretation of Numerical Results

In this paper we propose a very specific educational challenge that teachers can use to motivate ambitious and enthusiastic mathematics students who have mastered basic trigonometry and trig functions. The objective is to lead students to a result that will hopefully surprise and entertain and - more importantly - provide a lesson about when and when not to make inferences relating to "numerical error" when using mathematical software and calculators.

math.HO↗

arXiv2023

Tensor Completion with Provable Consistency and Fairness Guarantees for Recommender Systems

We introduce a new consistency-based approach for defining and solving nonnegative/positive matrix and tensor completion problems. The novelty of the framework is that instead of artificially making the problem well-posed in the form of an application-arbitrary optimization problem, e.g., minimizing a bulk structural measure such as rank or norm, we show that a single property/constraint: preserving unit-scale consistency, guarantees the existence of both a solution and, under relatively weak support assumptions, uniqueness. The framework and solution algorithms also generalize directly to tensors of arbitrary dimensions while maintaining computational complexity that is linear in problem size for fixed dimension d. In the context of recommender system (RS) applications, we prove that two reasonable properties that should be expected to hold for any solution to the RS problem are sufficient to permit uniqueness guarantees to be established within our framework. This is remarkable because it obviates the need for heuristic-based statistical or AI methods despite what appear to be distinctly human/subjective variables at the heart of the problem. Key theoretical contributions include a general unit-consistent tensor-completion framework with proofs of its properties, e.g., consensus-order and fairness, and algorithms with optimal runtime and space complexities, e.g., O(1) term-completion with preprocessing complexity that is linear in the number of known terms of the matrix/tensor. From a practical perspective, the seamless ability of the framework to generalize to exploit high-dimensional structural relationships among key state variables, e.g., user and product attributes, offers a means for extracting significantly more information than is possible for alternative methods that cannot generalize beyond direct user-product relationships.

cs.IR↗

arXiv2023

A Note on the Incompleteness of Gödel's Incompleteness Theorems

In this note we observe that automated theorem provers (ATPs) that recursively enumerate theorems in a particular way will fail to identify some valid theorems for a reason that is analogous to how Gödel proved the existence of what are now referred to as Gödel statements. This observation has no significant practical or theoretical implications, but it may be of pedagogical value for honing the intuition of students about recursive enumeration in the context of ATP.

math.GM↗

arXiv2023

Is Arrow's Dictator a Drinker?

We critique the formulation of Arrow's no-dictator condition to show that it does not correspond to the accepted informal/intuitive interpretation. This has implications for the theorem's scope of applicability.

econ.TH↗

arXiv2023

Partial Proof of a Conjecture with Implications for Spectral Majorization

In this paper we report on new results relating to a conjecture regarding properties of $n\times n$, $n\leq 6$, positive definite matrices. The conjecture has been proven for $n\leq 4$ using computer-assisted sum of squares (SoS) methods for proving polynomial nonnegativity. Based on these proven cases, we report on the recent identification of a new family of matrices with the property that their diagonals majorize their spectrum. We then present new results showing that this family can extended via Kronecker composition to $n>6$ while retaining the special majorization property. We conclude with general considerations on the future of computer-assisted and AI-based proofs.

cs.SC↗

arXiv2023

Metric Search for Rank List Compatibility Matching with Applications

As online dating has become more popular in the past few years, an efficient and effective algorithm to match users is needed. In this project, we proposed a new dating matching algorithm that uses Kendall-Tau distance to measure the similarity between users based on their ranking for items in a list. (e.g., their favourite sports, music, etc.) To increase the performance of the search process, we applied a tree-based searching structure, Cascading Metric Tree (CMT), on this metric. The tree is built on ranked lists from all the users; when a query target and a radius are provided, our algorithm can return users within the radius of the target. We tested the scaling of this searching method on a synthetic dataset by varying list length, population size, and query radius. We observed that the algorithm is able to query the best matching people for the user in a practical time, given reasonable parameters. We also provided potential future improvements that can be made to this algorithm based on the limitations. Finally, we offered more use cases of this search structure on Kendall-Tau distance and new insight into real-world applications of distance search structures.

cs.IR↗

arXiv2023

Choosing the Correct Generalized Inverse for the Numerical Solution of the Inverse Kinematics of Incommensurate Robotic Manipulators

Numerical methods for Inverse Kinematics (IK) employ iterative, linear approximations of the IK until the end-effector is brought from its initial pose to the desired final pose. These methods require the computation of the Jacobian of the Forward Kinematics (FK) and its inverse in the linear approximation of the IK. Despite all the successful implementations reported in the literature, Jacobian-based IK methods can still fail to preserve certain useful properties if an improper matrix inverse, e.g. Moore-Penrose (MP), is employed for incommensurate robotic systems. In this paper, we propose a systematic, robust and accurate numerical solution for the IK problem using the Mixed (MX) Generalized Inverse (GI) applied to any type of Jacobians (e.g., analytical, numerical or geometric) derived for any commensurate and incommensurate robot. This approach is robust to whether the system is under-determined (less than 6 DoF) or over-determined (more than 6 DoF). We investigate six robotics manipulators with various Degrees of Freedom (DoF) to demonstrate that commonly used GI's fail to guarantee the same system behaviors when the units are varied for incommensurate robotics manipulators. In addition, we evaluate the proposed methodology as a global IK solver and compare against well-known IK methods for redundant manipulators. Based on the experimental results, we conclude that the right choice of GI is crucial in preserving certain properties of the system (i.e. unit-consistency).

cs.RO↗

arXiv2023

Achieving Unit-Consistent Pseudo-Inverse-based Path-Planning for Redundant Incommensurate Robotic Manipulators

In this paper, we review and compare several velocity-level and acceleration-level Pseudo-Inverse-based Path Planning (PPP) and Pseudo-Inverse-based Repetitive Motion Planning (PRMP) schemes based on the kinematic model of robotic manipulators. We show that without unit consistency in the pseudo-inverse computation, path planning of incommensurate robotic manipulators will fail. Also, we investigated the robustness and noise tolerance of six PPP and PRMP schemes in the literature against various noise types (i.e. zero, constant, time-varying and random noises). We compared the simulated results using two redundant robotic manipulators: a 3DoF (2RP), and a 7DoF (2RP4R). These experimental results demonstrate that the improper Generalized Inverse (GI) with arbitrary selection of unit and/or in the presence of noise can lead to unexpected behavior of the robot, while producing wrong instantaneous outputs in the task space, which results in distortions and/or failures in the execution of the planned path. Finally, we propose and demonstrate the efficacy of the Mixed Inverse (MX) as the proper GI to achieve unit-consistency in path planning.

cs.RO↗

arXiv2023

Imposing Consistency Properties on Blackbox Systems with Applications to SVD-Based Recommender Systems

In this paper we discuss pre- and post-processing methods to induce desired consistency and/or invariance properties in blackbox systems, e.g., AI-based. We demonstrate our approach in the context of blackbox SVD-based matrix-completion methods commonly used in recommender system (RS) applications. We provide empirical results showing that enforcement of unit-consistency and shift-consistency, which have provable RS-relevant properties relating to robustness and fairness, also lead to improved performance according to generic RMSE and MAE performance metrics, irrespective of the initial chosen hyperparameter.

cs.IR↗

arXiv2023

An Admissible Shift-Consistent Method for Recommender Systems

In this paper, we propose a new constraint, called shift-consistency, for solving matrix/tensor completion problems in the context of recommender systems. Our method provably guarantees several key mathematical properties: (1) satisfies a recently established admissibility criterion for recommender systems; (2) satisfies a definition of fairness that eliminates a specific class of potential opportunities for users to maliciously influence system recommendations; and (3) offers robustness by exploiting provable uniqueness of missing-value imputation. We provide a rigorous mathematical description of the method, including its generalization from matrix to tensor form to permit representation and exploitation of complex structural relationships among sets of user and product attributes. We argue that our analysis suggests a structured means for defining latent-space projections that can permit provable performance properties to be established for machine learning methods.

cs.IR↗