Deep Zero Problem: Analysis & Learning

• Deep Zero Problem is a cross-disciplinary concept connecting holomorphic function uniqueness with neural network training challenges.
• It reveals that strict parity-based vanishing conditions in spaces like the Bargmann–Fock space determine whether interpolation and sampling are stable or degenerate.
• In neural networks, deep zero phenomena cause issues such as signal and rank collapse, with innovative initialization methods offering potential remedies.

The “Deep Zero Problem” encompasses several interconnected mathematical and algorithmic phenomena across functional analysis, neural network theory, and optimization. In its original form, the deep zero problem refers to the uniqueness properties of holomorphic functions subject to vanishing high-order derivatives (“jets”) at a small number of points, with related implications for interpolation and sampling. The terminology has migrated to applied mathematics and deep learning, denoting bottlenecks in signal propagation and optimization in very deep or high-dimensional models. Crucially, deep zero phenomena reveal rigid thresholds between uniqueness and breakdown of interpolation or sampling, with broad implications for function-theoretic operator theory, neural network initialization, optimization landscape analysis, and scalable learning protocols.

1. Formal Definition and Context

In analytic function spaces, the deep zero problem asks: given a reproducing-kernel Hilbert space $H$ of holomorphic functions (e.g., the Bargmann–Fock space $F(\mathbb{C})$), a finite set of centers $Z = \{z_1, ..., z_N\}$, and infinite subsets $E_k \subset \mathbb{N}_0$ for each center, does the condition

$$\forall k=1,\ldots,N,\;\forall j\in E_k,\;\; f^{(j)}(z_k) = 0$$

force $f \equiv 0$? Associated are interpolation (can prescribed jets be matched?) and sampling (is the norm controlled by the jets?) problems (Hedenmalm, 2022).

This vanishing condition is deeper (infinite order) than classic uniqueness theorems, and, for certain parity-based $E_1, E_2$, yields strikingly rigid answers: uniqueness holds under full parity constraints, but fails otherwise.

In modern deep learning, the “deep zero problem” also refers to failure modes in signal propagation or optimization—e.g., with all-zero or degenerate initialization leading to collapse of activation or gradient information, or rank-loss in Jacobians obstructing efficient learning (Zhao et al., 2021).

2. Deep Zero in the Bargmann–Fock Space

The central theoretical framework arises in the Fock–Bargmann space

$$F(\mathbb{C}) = \left\{ f:\; \text{entire}\; :\; \|f\|^2 = \int_\mathbb{C} |f(z)|^2 e^{-|z|^2} dA(z) < \infty \right\},$$

with reproducing kernel $K_w(z) = e^{z \bar w}$ and Fock translation $U_B f(z) = e^{-|B|^2/2 + z \bar B} f(z - B)$. The key uniqueness theorem asserts (Hedenmalm, 2022):

• If $E$ is all even (resp. odd) nonnegative integers and $f \in F(\mathbb{C})$ vanishes in derivatives $f^{(j)}(0)=0$ for $j\in E$ and $(U_B f)^{(j)}(0)=0$ for $j \notin E$, then $f \equiv 0$.

Where constraints are relaxed, nontrivial Gaussian-type functions can carry zeros without uniqueness; so full parity (even/odd) is necessary.

Interpolation and sampling fail to be bounded or well-conditioned in these settings; no uniform lower bound controls the norm via jets, reflecting intrinsic instability for “near-deep-zero” functions.

3. Group Symmetries and Proof Techniques

The deep zero uniqueness rests on group symmetries:

• Vanishing of all even jets at $0$ is equivalent to $f(z)$ being odd.
• Similarly, odd jets at $0$ and transformations at $B$ correspond to $f$ being an eigenfunction of projective representations (reflections and translations) within the rigid-motion group $z \mapsto p z - a$.
• Uniqueness is derived by showing this forces $f$ to be an eigenfunction for a translation $U_{-2B}$, which lacks point spectrum for $B \neq 0$, so necessarily $f = 0$.

By Bargmann’s isometry, the analytic setting maps to Fourier analysis on $L^2(\mathbb{R})$, with derivatives and translations expressed as modulation and reflection operators. The deep-zero seminorm can be reduced to

$$N(f)^2 = \sum_{j \in E} \frac{|f^{(j)}(0)|^2}{j!} + \sum_{j \notin E} \frac{|(U_B f)^{(j)}(0)|^2}{j!},$$

which becomes singular and loses control near the zero-set of certain special functions (e.g., $\cos(\beta t)$).

4. Operator-Theoretic and Algorithmic Aspects

In practical terms, deep zero problems translate to enforcing infinite linear systems on the Taylor coefficients of $f(z) = \sum a_n z^n$ (Hedenmalm, 2022). Finite truncations lead to large, ill-conditioned matrix systems, particularly near the zeros of controlling functions. Condition numbers of the resulting matrices grow rapidly, indicating severe sensitivity and instability—algorithmically, deep zeros encode a form of “intrinsic ill-conditioning” in the underlying system.

In the operator-theoretic view, deep zeros correspond to the joint kernel of families of differential and translation operators, with implications for the structure of invariant subspaces in $H$.

5. Connections to Time–Frequency Analysis and the HRT Conjecture

Recent work extends the deep zero problem to links with the Heil–Ramanathan–Topiwala (HRT) conjecture in time-frequency analysis (Li et al., 14 Jan 2026). Under the Bargmann transform,

• Weyl–shift operators $U_\beta$ map to time–frequency shifts in $L^2(\mathbb{R})$.
• Fock-space deep-zero conditions (e.g., vanishing of $f^{(j)}$ in congruence classes at different points) can be recast as linear dependence constraints among families of Weyl shifts $U_{\lambda_k} f$.
• If the HRT conjecture holds for specific configurations (roots of unity), then the only solution to the associated deep-zero system is $f \equiv 0$.

This connection resolves instances of a generalized deep zero problem for $d = 2, 3, 4, 6$, with open questions at higher orders.

6. Deep Zero Phenomena in Neural Network Theory

The “deep zero problem” label has been adopted in analyzing initialization and optimization bottlenecks in deep neural networks (Zhao et al., 2021, Chen et al., 19 Feb 2025):

• All-zero weight initialization leads to symmetry and gradient collapse, stalling training (forward-signal collapse).
• Even structured (identity-only) initializations induce “rank collapse” in hidden activations: learning trajectories are confined to low-dimension subspaces, restricting expressivity.
• Hadamard transform-based ZerO initialization circumvents these degeneracies—preserving dynamical isometry, preventing expressivity collapse, supporting ultra-deep stable training without batch-normalization, and yielding reproducible, low-rank, sparse solutions.

Overparameterized networks satisfy sufficient conditions for zero training loss on generic data—explicit minimizers exist independent of optimization method (see (Chen et al., 19 Feb 2025) for constructive algorithms). However, increasing depth not width can degrade conditioning, due to proliferation of “rank-loss” regions where the Jacobian drops rank, impeding gradient-based convergence.

In serial deep networks, nonzero duality gap persists for depth $\geq 3$; architectures employing parallel branches and $L$-th power regularization regain strong duality and admit convex reformulations (Wang et al., 2021).

7. Applications, Open Problems, and Future Directions

Deep zero phenomena have broad mathematical and practical impact:

• Extensions to multi-center deep zeros, analysis in Bergman, Paley–Wiener, de Branges–Rovnyak spaces.
• Quantitative stability: characterizing how close near-solutions are to true zeros, especially in systems perturbed by data or implementation noise.
• Discrete-time/frequency analogs and finite-dimensional tests for deep zero properties.
• Interplay with invariant subspaces and vanishing moments, as seen in wavelet and Gabor analysis.
• Operator-theoretic characterization of the closure and structure of joint kernels.

Open questions include:

• For which combinations of centers and jet orders does uniqueness persist?
• Can time–frequency independence be fully classified via the HRT conjecture for all roots of unity?
• In neural networks, can explicit detection and amelioration of “deep zero” bottlenecks yield provably faster or more robust training regimes?

The deep zero problem thus constitutes a cross-disciplinary concept tying together analytic uniqueness, optimization landscape pathology, functional rigidity, and design principles for scalable, stable machine learning and operator representation systems. The ongoing expansion of its theory promises further unification of complex analysis, time–frequency mathematics, and deep learning dynamics.