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Thinking Summary

**Describing GeGLU Geometry** I'm focusing on the geometry of the GeGLU activation function. It involves the product of a GELU-transformed gate and a linear value. In a scalar case, I can express it as f(a, b) = GELU(a) * b, which helps visualize it in two dimensions. In terms of gradients, the gating creates an oblique hyperplane, smoothly transitioning between 0 and 1. I think it’s helpful to consider this piecewise and note how the GEGLU behaves like a cubic on the negative side near zero, resulting in a geometric shape that's not entirely symmetrical. **Defining GeGLU Geometry** I'm planning to provide a clear definition and illustrate the shape in 2D, focusing on the gating dimension versus the scaling dimension. I'll also discuss the gradient vector field and explain the iso-lines, which resemble radial "sausages" oriented along the value dimension. Additionally, I want to highlight the differentiation between the gating axis, referred to as the "gate dimension," and the "value dimension." It's important to deliver this in an organized way that clearly conveys the concepts.

Geometry of a single-neuron GeGLU
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(Write x ⁣= ⁣(xg,xv)x\!=\!(x_g,\,x_v), with xgx_g the gate component and xvx_v the value component that will ultimately flow forward.)

Scalar form:
fGeGLU(xg,xv)=GELU(xg)    xv  =xgΦ(xg)  xvf_{\text{GeGLU}}(x_g, x_v)=\text{GELU}(x_g)\;\cdot\;x_v   =x_g\,\Phi(x_g)\;x_v

  1. Domain & codomain
    • Domain: the 2-D plane (xg,xv)R2(x_g,x_v)\in\mathbb R^2.
    • Codomain: the real line R\mathbb R (for one neuron) or, in the vector case, the dff/2d_{ff}/2-dimensional subspace spanned by the “value” projection.
    Geometrically we are mapping each point of the plane to a signed height above/below it.

  2. Separating axes
    • The xgx_g-axis controls a smooth gate, ranging continuously from 0 (far negative) through 0.5 (at the origin) toward 1 (large positive).
    • The xvx_v-axis is passed through unchanged and merely gets scaled by that gate.
    Thus the plane is foliated into vertical lines (constant xgx_g) that are uniformly compressed or expanded.

  3. Surface shape
    Plotting z=fGeGLU(xg,xv)z=f_{\text{GeGLU}}(x_g,x_v) gives a saddle-like surface that is:

    • Flat along the entire xgx_g axis at xv ⁣= ⁣0x_v\!=\!0 (because the value is zero).
    • As xv|x_v| increases the surface “fans out’’ in the ±xv\pm x_v direction, but its amplitude is modulated by the bell-shaped factor xgΦ(xg)x_g\Phi(x_g) (Fig. below).

    For fixed xvx_v the cross-section is exactly the 1-D GELU curve scaled by xvx_v; for fixed xgx_g it is a straight line through the origin with slope GELU(xg)\text{GELU}(x_g).

  4. Level sets (iso-contours)
    Solving f(xg,xv)=cf(x_g,x_v)=c gives
    xv=c/GELU(xg)x_v = c / \text{GELU}(x_g) whenever GELU(xg)0\text{GELU}(x_g)\neq0.
    These are rational curves that resemble hyperbolas but whose arms pinch smoothly toward the xg<0x_g<0 region where the gate shrinks to zero. No contour ever crosses the negative xgx_g axis because the output there is arbitrarily close to 0.

  5. Gradient field
    f=(xvGELU(xg),  GELU(xg))\nabla f = \bigl(x_v\,\text{GELU}'(x_g),\; \text{GELU}(x_g)\bigr)
    • Along the xvx_v direction gradients are constant w.r.t. xvx_v: each vertical line has the same slope equal to GELU(xg)\text{GELU}(x_g).
    • Gradients vanish as xg ⁣ ⁣x_g\!\to\!-\infty (saturation), peak around xg ⁣ ⁣1x_g\!\approx\!1, and decay again for very large xgx_g due to the xgΦ(xg)x_g\Phi(x_g) factor plateauing at xgx_g.

Vectorised GeGLU inside a Transformer FFN
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Let the input vector hRdh\in\mathbb R^{d} be linearly projected into two halves:

g=hWgRdff/2g = hW_g\in\mathbb R^{d_{ff}/2} (gate)
v=hWvRdff/2v = hW_v\in\mathbb R^{d_{ff}/2} (value)

and produce y=GELU(g)  vy = \text{GELU}(g)\ \otimes\ v. (Hadamard product.)

Geometrically:

  1. The value subspace spanned by columns of WvW_v forms a (dff/2)(d_{ff}/2)-dimensional affine subspace of the model’s representation space.
  2. The gate vector GELU(g)[0,)dff/2\text{GELU}(g)\in[0,\infty)^{d_{ff}/2} acts as a smooth, element-wise scaling mask that compresses coordinates of that subspace toward the origin depending on how “positive’’ the companion coordinates in gg are.
  3. Consequently, the reachable set of yy is a warped, axis-aligned “cone’’ inside that subspace whose opening width varies smoothly with gg. No new directions are introduced; directions are re-weighted.

Comparisons & intuition
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ReLU gate: sharp half-space clipping, output lies on two flat half-cones separated by a hard plane xg=0x_g=0.
GELU gate: replaces that plane by a sigmoidal transition band, bending the cone walls smoothly. The resulting surface is everywhere differentiable, giving GeGLU its smoother optimization landscape while still preserving a gating geometry (outputs shrink continuously to zero as any gate coordinate drifts negative).

Bottom line
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GeGLU carves the (g,v)(g,v) plane into smoothly-gated linear pencils: each “value’’ axis is a straight line whose amplitude envelope is the soft bell curve of the GELU applied to its paired “gate’’ axis. In higher dimension, this becomes an axis-aligned, differentiable gating cone embedded in the FFN’s hidden space.

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