The Inverse Problem: Why Source Estimation Is Hard

More unknowns than equations

Take the forward model \mathbf{v}=\mathbf{L}\mathbf{j} and try to solve for \mathbf{j}. A high-density EEG cap has perhaps 64–256 sensors; a distributed source space has thousands of dipoles, each with up to three orientation components. So \mathbf{L} is a very wide matrix and the system is massively underdetermined — there are far more knobs than constraints.

\mathbf{v} = \mathbf{L}\mathbf{j},\qquad \mathbf{L}\in\mathbb{R}^{M\times 3N},\quad M \ll 3N

The inverse problem is underdetermined: far fewer sensors M than source unknowns 3N. Infinitely many current distributions j reproduce the same measurement v.

The core headache of source localization: you have far fewer sensors than brain locations to explain, so the equations can't pin down a unique answer. Infinitely many different current patterns produce exactly the same scalp recording.

M: Number of sensors — at most a few hundred.
3N: Number of source unknowns — often thousands.
M \ll 3N: Far more unknowns than measurements — the problem is underdetermined.
\mathbf{L}: The lead-field matrix (short and wide).

Concretely, \mathbf{L} has a large null space: any current pattern \mathbf{j}_0 with \mathbf{L}\mathbf{j}_0=\mathbf{0} — a so-called silent source producing no scalp field whatsoever — can be added to any solution without changing a single measurement. Helmholtz proved in the 19th century that the general inverse problem has no unique solution; this is not a numerical nuisance but a fact of physics you cannot compute your way out of.

Ill-posed, not just underdetermined

Even setting non-uniqueness aside, the inverse is ill-posed in Hadamard's sense: the solution does not depend continuously on the data. Because volume conduction is a smoother (a spatial low-pass filter), inverting it is a de-smoothing operation that violently amplifies noise. Tiny measurement errors — a bit of EMG, a reference drift, a single bad channel — can map to wildly different, physiologically absurd source estimates.

Escape 1 — fit a few dipoles

The oldest strategy sidesteps the null space by drastically shrinking the unknowns: assume the data were generated by just one or a few equivalent current dipoles, and fit their positions and moments by nonlinear least squares.

\hat{\boldsymbol{\theta}} = \arg\min_{\boldsymbol{\theta}}\;\bigl\lVert \mathbf{v} - \mathbf{L}(\boldsymbol{\theta})\,\mathbf{q}\bigr\rVert^{2}

Equivalent-dipole fitting. θ collects the dipole positions (which enter L nonlinearly); given the positions, the moments q are linear and solved in closed form.

One escape route: assume just a handful of dipoles and hunt for the positions that best reproduce the data. The positions enter awkwardly (nonlinearly), but once you fix them, finding each dipole's strength is an easy linear step.

\boldsymbol{\theta}: The dipole positions, searched over — they enter \mathbf{L} nonlinearly.
\mathbf{q}: The dipole moments, solved in closed form once positions are fixed.
\mathbf{v}-\mathbf{L}(\boldsymbol{\theta})\mathbf{q}: The mismatch between measured and predicted data.
\arg\min: Pick the parameters that make the mismatch smallest.

With few dipoles the problem becomes overdetermined and well-behaved — but only if the assumption is true. Choose too few dipoles and you miss real sources; choose too many and the fit turns unstable and multimodal, with the optimizer trapped in local minima. Dipole fitting shines for focal, well-understood generators (an early sensory evoked response, an epileptic spike) and fails for spatially distributed cognitive activity.

Escape 2 — keep all sources, add a prior

The modern strategy keeps the full distributed source space and resolves the ambiguity by choosing the solution that best satisfies a prior — a preference for, say, the smallest-energy current distribution, or the smoothest one. This is the domain of the minimum-norm estimate and its relatives, the subject of the next guide.

\hat{\mathbf{j}} = \arg\min_{\mathbf{j}}\;\underbrace{\lVert \mathbf{v}-\mathbf{L}\mathbf{j}\rVert^{2}}_{\text{fit the data}} \;+\; \lambda\,\underbrace{R(\mathbf{j})}_{\text{obey the prior}}

The regularized inverse: trade data fit against a penalty R(j) encoding prior belief, balanced by λ. Different priors give MNE, LORETA, beamformers, and sparse solvers.

The other escape: keep all sources but add a 'prior' — a preference for solutions that behave sensibly. You balance fitting the data against obeying that preference, with \lambda setting the trade-off. Which prior you pick gives MNE, LORETA, beamformers, and so on.

\lVert\mathbf{v}-\mathbf{L}\mathbf{j}\rVert^{2}: The data-fit term — how well the sources explain the recording.
R(\mathbf{j}): The penalty encoding your prior belief about the source.
\lambda: The knob balancing data fit against the prior.

What 'solving' honestly means

Here is the uncomfortable truth to internalize before trusting any source map: you never recover the true current distribution. You recover the element of the solution set that your prior likes best. Change the prior — minimum energy vs. maximum smoothness vs. sparsity — and the picture changes, because the true source and the null-space component are, by construction, indistinguishable from the data alone. Regularization does not reveal truth; it selects a preference. See also the resolution matrix, which makes this precise.