We thank all the reviewers for their detailed comments. Related work: We will include comparison with [Fiat et al., 2019] in the main section. Most analysis of DNNs with ReLU is on what happens at initialisation. In a DNN with ReLU, NPV and NPF are not statistically independent at initialisation, i.e., Assumption 5.1 does not Hence, though Assumption 5.1 may not hold However, we will move relevant work (example [Srivastava et al., However, studying these in our framework is future work. MNIST and CIFAR-10 are used as standard datasets in most analytical works such as ours, see [Arora et al., 2019] for By generalisation we mean performance on test data.

Bayesian experimental design (BED) provides a principled framework for optimizing data collection, but existing approaches do not apply to crucial real-world settings such as dynamical systems with partial observability, where only noisy and incomplete observations are available. These systems are naturally modeled as state-space models (SSMs), where latent states mediate the link between parameters and data, making the likelihood -- and thus information-theoretic objectives like the expected information gain (EIG) -- intractable. In addition, the dynamical nature of the system requires online algorithms that update posterior distributions and select designs sequentially in a computationally efficient manner. We address these challenges by deriving new estimators of the EIG and its gradient that explicitly marginalize latent states, enabling scalable stochastic optimization in nonlinear SSMs. Our approach leverages nested particle filters (NPFs) for efficient online inference with convergence guarantees. Applications to realistic models, such as the susceptible-infected-recovered (SIR) and a moving source location task, show that our framework successfully handles both partial observability and online computation.

A deep neural network (DNN) with ReLU activations has many gates, and the on/off status of each gate changes across input examples as well as network weights. For a given input example, only a subset of gates are active, i.e., on, and the sub-network of weights connected to these active gates is responsible for producing

We thank all the reviewers for their detailed comments. Related work: We will include comparison with [Fiat et al., 2019] in the main section. Most analysis of DNNs with ReLU is on what happens at initialisation. In a DNN with ReLU, NPV and NPF are not statistically independent at initialisation, i.e., Assumption 5.1 does not Hence, though Assumption 5.1 may not hold However, we will move relevant work (example [Srivastava et al., However, studying these in our framework is future work. MNIST and CIFAR-10 are used as standard datasets in most analytical works such as ours, see [Arora et al., 2019] for By generalisation we mean performance on test data.

However, current sequencing-based technologies suffer from low spatial resolution, and substantial inter-tissue variability in protein expression further compromises the performance of existing molecular data prediction methods. In this work, we introduce the novel task of spatial super-resolution for sequencing-based spatial proteomics (seq-SP) and, to the best of our knowledge, propose the first deep learning model for this task--Neural Proteomics Fields (NPF). NPF formulates seq-SP as a protein reconstruction problem in continuous space by training a dedicated network for each tissue. The model comprises a Spatial Modeling Module, which learns tissue-specific protein spatial distributions, and a Morphology Modeling Module, which extracts tissue-specific morphological features. Furthermore, to facilitate rigorous evaluation, we establish an open-source benchmark dataset, Pseudo-Visium SP, for this task. Experimental results demonstrate that NPF achieves state-of-the-art performance with fewer learnable parameters, underscoring its potential for advancing spatial proteomics research. Our code and dataset are publicly available at https://github.com/Bokai-Zhao/NPF.

I have increased my score. Original summary: The authors define two key properties of a ReLU DNN: the Neural Path Feature (NPF) and the Neural Path Value (NPV). The NPF encodes which paths are active and the input features associated with those paths.

Rectified linear unit (ReLU) activations can also be thought of as \emph{gates}, which, either pass or stop their pre-activation input when they are \emph{on} (when the pre-activation input is positive) or \emph{off} (when the pre-activation input is negative) respectively. A deep neural network (DNN) with ReLU activations has many gates, and the {on/off} status of each gate changes across input examples as well as network weights. For a given input example, only a subset of gates are \emph{active}, i.e., on, and the sub-network of weights connected to these active gates is responsible for producing the output. At randomised initialisation, the active sub-network corresponding to a given input example is random. During training, as the weights are learnt, the active sub-networks are also learnt, and potentially hold very valuable information. In this paper, we analytically characterise the role of active sub-networks in deep learning. To this end, we encode the {on/off} state of the gates of a given input in a novel \emph{neural path feature} (NPF), and the weights of the DNN are encoded in a novel \emph{neural path value} (NPV). Further, we show that the output of network is indeed the inner product of NPF and NPV. The main result of the paper shows that the \emph{neural path kernel} associated with the NPF is a fundamental quantity that characterises the information stored in the gates of a DNN. We show via experiments (on MNIST and CIFAR-10) that in standard DNNs with ReLU activations NPFs are learnt during training and such learning is key for generalisation. Furthermore, NPFs and NPVs can be learnt in two separate networks and such learning also generalises well in experiments.