PIN v4: A Shared Coordinate System for Model Families including Compression, Design by Specification and Inference Cost

.A neural network's weights can be tied together so that many of them read the same stored value. This paper reports what that buys, measured rather than argued. On tiny-imagenet-200, scored on the official validation split against a dense model trained for three hundred epochs with augmentation, a folded model matches it at thirty times less storage and beats it at eight. On CIFAR-10 a model holding seventy-three stored values scores 0.478 where an unconstrained network holding 2,621,440 reaches 0.421. In a transformer, 25,720 values score 0.420 against 461,760 weights scoring 0.356 — smaller and better, because a model of that size on that data overfits and the constraint prevents it.

The mechanism is not compression. Tying two weights means the model can no longer tell those inputs apart, so a fold is an imposed blind spot, and it pays when it is blind to something the task does not need. On that reading a convolution and a vision transformer's patch embedding are one construction at different settings of two numbers; both are partitions; and every such blindness beats an arbitrary partition holding identical values by five to eleven standard deviations. 

The stored values and the partition are separate objects. Forty-five models can share one body, holding 6,354 values against 107,730 stored individually for an accuracy difference of 0.001, and one set of values can be read through several partitions at once, so an ensemble arrive at single-model storage.

Storage and arithmetic are separate currencies too. An arbitrary tying is small and still costs a full dense product, because every weight is an independent lookup; a convolution holding the same 433 values uses 28 times fewer multiplies and is 0.13 more accurate. The same constraint reduces what must be stored and exposes structure that reduces what must be computed.

These combine into a design procedure: calibrating a storage curve once predicts what will be built to a median of two thousandths across a sixfold range, and carries a confidence band derived from its own local slope.

Version 1.1's constant-depth claim contained a codimension error and is withdrawn; a corrected form survives and is stronger. Everything here is subject to change and later versions supersede.