Reproducibility in Matrix and Tensor Decompositions: Focus on Model Match, Interpretability, and Uniqueness

Consider a region consisting of several individual conductors with a definite charge density in the region between these conductors. The second uniqueness theorem states that if the total charge on each conductor and the charge density in the in-between region are known, then the electric field can be uniquely determined.
In contrast, consider that the electric field is non-unique and apply Gauss's law in divergence form in the region between the conductors and the integral form to the...

The concept of the inertia tensor is employed to depict the mass distribution and rotational inertia of a solid or rigid object. This tensor is expressed through a three-by-three matrix. Each component within this matrix corresponds to varying moments of inertia about specific axes.
The diagonal components of the inertia tensor matrix represent the moments of inertia concerning the principal axes of the object. These primary axes are defined as the axes where the object experiences the least...

Base complementarity between the three base pairs of mRNA codon and the tRNA anticodon is not a failsafe mechanism. Inaccuracies can range from a single mismatch to no correct base pairing at all. The free energy difference between the correct and nearly correct base pairs can be as small as 3 kcal/ mol. With complementarity being the only proofreading step, the estimated error frequency would be one wrong amino acid in every 100 amino acids incorporated. However, error frequencies observed in...

The extracellular matrix or ECM holds cells together to form a tissue and allows the cells within the tissue to communicate. ECM comprises proteins such as fibronectin, collagen, laminin, etc. The most abundant protein in this space is collagen. Collagen fibers are interwoven with carbohydrate-containing protein molecules called proteoglycans. ECM allows cell migration and provides a structural scaffold at cell adhesion that anchors the cell when the extracellular matrix proteins interact with...

Matrix metalloproteases (MMPs) are enzymes involved in the hydrolysis of proteins and glycoproteins of the extracellular matrix. MMPs are essential for the migration and proliferation of cells through the dense matrix network, throughout embryonic development, and throughout morphogenesis. The first MMP activity discovered was a collagenase in a tadpole's tail undergoing metamorphosis. The active collagen deposition and modifications lead to the morphogenesis of tadpoles into the adult...

Proteoglycans are extensively glycosylated proteins, commonly found in the extracellular matrix, interwoven with collagen fibers. Hyaline cartilage, the most common type of cartilage in the body, consists of short and dispersed collagen fibers associated with large amounts of proteoglycans. These proteoglycans have long negative charges that attract cations, which in turn attract water molecules. This influx of ions and water molecules swells up the proteoglycan like a water-soaked gel that can...