One interesting thing to notice is how you can remodel xor into being a linear function by using u + v as input 1 and u * v as input 2 which means it can be represented in a NN without a hidden layer. And not only xor, it keeps all other logic gates simple. So only by transforming inputs one can reduce network complexity. Perhaps a field ripe for research.
indeed,
there is an extensive work done in kernel learning that is facinating
and one of the applications that still do these transformations are satellite imagery/multispectral imagery, you can get more information just from calculating the ndvi from the different bands of your image, which make it easy for your models to make decisions
I misread this as if "there is no non-linearity". there is still non-linearity, it is just renamed and reshuffled into new operators. basically renaming apples into oranges.
Well, it's more like fruits and vegetables. The author proposed a normalized inner product as replacement for the standard inner product.
It's not an activation function, because it has the learnable weights of a linear projection (mat vec multiplication) and the clamping properties of an activation function all in one.
My personal issue with the proposal is that it essentially doubles the amount of memory needed on-chip.
Yat-Product GEMMV now needs to store the running total of the inner product and the norm of the input vectors. That's a big cost increase for something that might not improve performance all that much.
that's a great point you made, but the goal of this research paper is not to improve the performance, but to show that you can train deep neural networks without the need of activation functions, normalization layers, deep neural networks.
one simple usecase for them is physics-informed neural networks and neural ODEs, where using activation functions is discouraged, mainly because they aren't infinitly differentiable, and they use the tanh or the sin most of the time,
this kernel i introduced works better then the neurons followed with a tanh to solve different PDEs
basicly the real "non-linearity" in deep learning have always been the orthogonality, squashing functions make it easy for the neurons to tap into the orthogonality, while most of the activation functions "lie" about their orthogonality by setting the dot product score to "0", and a dot product of 0 between two vectors means they are orthogonal (linear indep)
what i did was rely on both the angular information and spatial information between the input x and the weight w to measure how "similar" they are.
the lower bound of the yat-product is 0, and it is achieved only when two vectors are orthogonal and away
I was able to create a new kernel that allows you to learn non-linearity without using activation functions, making the models whitebox, and without any information loss.
they are one of the reasons neural networks are blackbox,
we lose information about the data manifold the deeper we go in the network, making it impossible to trace back the output
this preprint is not coming from a standpoint of optimizing the inference/compute, but from trying to create models that we can interpret in the future and control
Less information loss -> Less params? Please correct me if I got this wrong. The Intro claims:
"The dot product itself is a geometrically impoverished measure, primarily capturing alignment while conflating magnitude with direction and often
obscuring more complex structural and spatial relationships [10, 11, 4, 61, 17]. Furthermore, the way current activation functions achieve non-linearity can exacerbate this issue. For instance, ReLU (f (x) = max(0, x)) maps all negative pre-activations, which can signify a spectrum of relationships from weak dissimilarity to strong anti-alignment, to a single zero output. This thresholding, while promoting sparsity, means the network treats diverse inputs as uniformly orthogonal or linearly independent for onward signal propagation. Such a coarse-graining of geometric relationships leads to a tangible loss of information regarding the degree and nature of anti-alignment or other neg-
ative linear dependencies. This information loss, coupled with the inherent limitations of the dot product, highlights a fundamental challenge."
yes,
since you can learn to represent the same problem with less amount of params,
however most of the architectures are optimized for the linear product, so we gotta figure out a new architecture for it
