Where does gradient collide?

The vector built from a scalar field's partial derivatives, pointing in the direction of steepest increase, with length equal to that steepest rate of change. In vector calculus, as set out in Marsden and Tromba's Vector Calculus, the gradient of a differentiable function f from Rⁿ to R is the field grad f = (∂f/∂x₁, ..., ∂f/∂x_n); at each point it names the direction to move for the fastest rise in f. The referent is a mathematical object, a field of vectors defined over a domain, and the unit is whatever the function measures per unit of position. It shares nothing with the health meaning but the word: the object is a derivative, it carries no data and no noise, and it is studied for its own structure.[1]

Unit

Change in the field per unit of position (a vector)

Recovery means

Not applicable

Measured by

Computed analytically as the partial derivatives of a differentiable scalar field

The same vector of partial derivatives, but taken of a loss function over a model's parameters and used as the direction to step against during optimization. Gradient descent, whose form Cauchy set out in 1847 and which modern machine learning runs at scale, updates each parameter by subtracting a small multiple of the loss gradient, so the model moves downhill toward lower error. Rumelhart, Hinton, and Williams showed in 1986 that backpropagation computes this gradient efficiently through a layered network. The referent is an optimization procedure, and the gradient here is a means to an end, the signal that tells the algorithm which way to move, not the mathematical object contemplated for its own sake.[2]

Unit

Change in the loss per unit change in a parameter

Recovery means

Not applicable

Measured by

Computed by backpropagation over a network's parameters at each training step

A spatial concentration profile of a signalling molecule, a morphogen, that tells cells where they sit and so what to become. Wolpert's 1969 account of positional information, the French flag model, proposed that a graded morphogen concentration across a tissue gives each cell a coordinate, which it reads against thresholds to choose a fate. The referent is a physical field of molecules across an embryo, measured as concentration per unit distance, and the gradient does causal work in the tissue rather than describing the rate of change of some other quantity. It is a thing in the world, a distribution of matter, not an operator applied to a function and not a slope fitted to data.[3]

Unit

Morphogen concentration per unit distance across tissue

Recovery means

Not applicable

Measured by

Assayed as spatial concentration profiles of morphogens across a developing tissue

A stepwise relation between socioeconomic position and health, in which each step down the social hierarchy carries worse health than the step above, not merely a gap between rich and poor. Marmot and colleagues' Whitehall studies of British civil servants found this gradient across employment grades: mortality and morbidity rose at every lower rank, including among the well paid and securely employed. The referent is a population-level association between rank and outcome, measured as a slope of health across social position. Here gradient means a statistical relationship in epidemiological data, not a vector, not an optimization step, and not a molecule; only the bare idea of a graded change survives the crossing.[4]

Unit

Change in a health outcome per step of social position

Recovery means

Not applicable

Measured by

Estimated as a slope of health outcomes across ranked socioeconomic categories