RBF Layer — Machine Learning/Neural Networks

RBF Layer

Machine Learning / Neural Networks

A radial basis function network with its fitted parameters baked in: each centre answers φᵢ = exp(−‖u − cᵢ‖² / (2σᵢ²)), and the readout is yₖ = Σᵢ Wₖᵢ·φᵢ + bₖ.

The network engineers reach for when they have to defend the model: a centre only responds near its own neighbourhood, so what the network does in one region of the input space cannot be moved by a centre fitted to another – and every parameter is a place, a width or a weight rather than an uninterpretable coefficient. It is also the one model here whose parameters can be fitted with no framework at all: k-means for the centres, least squares for the readout.

Ports

• u – the feature column, [m,1], where m is the NUMBER OF COLUMNS of Centres. A different size stops the run rather than broadcasting.
• Output – y, a column [p,1], where p is the NUMBER OF ROWS of Output Weights. With one row this is the plain scalar RBF network.

Parameters

• Centres – [N,m], one centre per ROW. That is k-means' orientation and the one K Means Assign uses in this library, so a fitted set pastes into either block unchanged. Its shape sets both the number of centres and the required input width.
• Widths – σ, either one per centre ([N,1]) or a single scalar shared by all of them. Every value must be > 0. A width is a DISTANCE in the same units as the input, so it scales with the features rather than with the output.
• Output Weights – [p,N], the linear readout: one row per output, one column per centre. Its row count is the output height.
• Bias – [p,1], added after the readout. This is what the network returns far away from every centre, where all φ have decayed to zero – so it is the model's default answer, not a cosmetic offset.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. Every parameter is structural and baked into the body; there is no tunable parameter, and the network is fully unrolled, so no backend contains a loop or an index computed at run time.

The widths are folded once, at config load, into 1/(2σ²) – so the emitted body multiplies where the formula divides, and no target carries a division per centre per sample.

The three HDL targets are simulation-only real arithmetic, quantizing only at the port boundary, because a Gaussian needs exp and that does not belong in a Q16.16 datapath. They simulate correctly and are not offered for synthesis – the same choice Dense Layer makes for its Tanh and Sigmoid activations. PLC Structured Text needs no such caveat: EXP is part of IEC 61131-3.

Simulink bridge

None. Simulink's Deep Learning blocks take a trained network object, which no parameter mapping can carry, and there is no RBF block in the base product to map onto. The bridge reports the block rather than dropping it silently, and it has no parity testbench, which is the documented consequence of Support::None rather than a gap.

Notes

• Algebraic and stateless: the output depends only on the current input, so the block cannot break an algebraic loop.
• Nonlinear, and deliberately carries no state space – the Gaussian is not a linear form and a fabricated one would let model reduction merge away exactly the nonlinearity the block exists for.
• It cannot overflow. The exponent argument is −‖u − c‖²·(1/(2σ²)), which is never positive, so exp is only ever evaluated on [−∞, 0] and saturates to zero rather than to infinity. Far from every centre the block returns its Bias, which is the honest answer for a local model asked about a place it was never fitted on.
• The squared distance is used as-is – there is no square root anywhere, because the exponent takes the squared distance directly.