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Text generation with Large Language Models (LLMs) is known to be memory bound due to the combination of their auto-regressive nature, huge parameter counts, and limited memory bandwidths, often resulting in low token rates. Speculative decoding has been proposed as a solution for LLM inference acceleration. However, since draft models are often unavailable in the modern open-source LLM families, e.g., for Llama 2 7B, training a high-quality draft model is required to enable inference acceleration via speculative decoding. In this paper, we propose a simple draft model training framework for direct alignment to chat-capable target models. With the proposed framework, we train Llama 2 Chat Drafter 115M, a draft model for Llama 2 Chat 7B or larger, with only 1.64\% of the original size. Our training framework only consists of pretraining, distillation dataset generation, and finetuning with knowledge distillation, with no additional alignment procedure. For the finetuning step, we use instruction-response pairs generated by target model for distillation in plausible data distribution, and propose a new Total Variation Distance++ (TVD++) loss that incorporates variance reduction techniques inspired from the policy gradient method in reinforcement learning. Our empirical results show that Llama 2 Chat Drafter 115M with speculative decoding achieves up to 2.3 block efficiency and 2.4$\times$ speed-up relative to autoregressive decoding on various tasks with no further task-specific fine-tuning., Comment: 8 pages, 3 figures, Published at the ICLR 2024 Workshop on Understanding of Foundation Models (ME-FoMo)

Although the rise of Large Language Models (LLMs) in enterprise settings brings new opportunities and capabilities, it also brings challenges, such as the risk of generating inappropriate, biased, or misleading content that violates regulations and can have legal concerns. To alleviate this, we present "LLMGuard", a tool that monitors user interactions with an LLM application and flags content against specific behaviours or conversation topics. To do this robustly, LLMGuard employs an ensemble of detectors., Comment: accepted in demonstration track of AAAI-24

Food touches our lives through various endeavors, including flavor, nourishment, health, and sustainability. Recipes are cultural capsules transmitted across generations via unstructured text. Automated protocols for recognizing named entities, the building blocks of recipe text, are of immense value for various applications ranging from information extraction to novel recipe generation. Named entity recognition is a technique for extracting information from unstructured or semi-structured data with known labels. Starting with manually-annotated data of 6,611 ingredient phrases, we created an augmented dataset of 26,445 phrases cumulatively. Simultaneously, we systematically cleaned and analyzed ingredient phrases from RecipeDB, the gold-standard recipe data repository, and annotated them using the Stanford NER. Based on the analysis, we sampled a subset of 88,526 phrases using a clustering-based approach while preserving the diversity to create the machine-annotated dataset. A thorough investigation of NER approaches on these three datasets involving statistical, fine-tuning of deep learning-based language models and few-shot prompting on large language models (LLMs) provides deep insights. We conclude that few-shot prompting on LLMs has abysmal performance, whereas the fine-tuned spaCy-transformer emerges as the best model with macro-F1 scores of 95.9%, 96.04%, and 95.71% for the manually-annotated, augmented, and machine-annotated datasets, respectively., Comment: 13 pages, 6 main figures and 2 in appendices, and 3 main tables; Accepted for publication in LREC-COLING 2024

We study a contract design problem between a principal and multiple agents. Each agent participates in an independent task with binary outcomes (success or failure), in which it may exert costly effort towards improving its probability of success, and the principal has a fixed budget which it can use to provide outcome-dependent rewards to the agents. Crucially, we assume the principal cares only about maximizing the agents' probabilities of success, not how much of the budget it expends. We first show that a contract is optimal for some objective if and only if it is a successful-get-everything contract. An immediate consequence of this result is that piece-rate contracts and bonus-pool contracts are never optimal in this setting. We then show that for any objective, there is an optimal priority-based weighted contract, which assigns positive weights and priority levels to the agents, and splits the budget among the highest-priority successful agents, with each such agent receiving a fraction of the budget proportional to her weight. This result provides a significant reduction in the dimensionality of the principal's optimal contract design problem and gives an interpretable and easily implementable optimal contract. Finally, we discuss an application of our results to the design of optimal contracts with two agents and quadratic costs. In this context, we find that the optimal contract assigns a higher weight to the agent whose success it values more, irrespective of the heterogeneity in the agents' cost parameters. This suggests that the structure of the optimal contract depends primarily on the bias in the principal's objective and is, to some extent, robust to the heterogeneity in the agents' cost functions., Comment: 26 pages

Speculative decoding is an inference-acceleration method for large language models (LLMs) where a small language model generates a draft-token sequence which is further verified by the target LLM in parallel. Recent works have advanced this method by establishing a draft-token tree, achieving superior performance over a single-sequence speculative decoding. However, those works independently generate tokens at each level of the tree, not leveraging the tree's entire diversifiability. Besides, their empirical superiority has been shown for fixed length of sequences, implicitly granting more computational resource to LLM for the tree-based methods. None of the existing works has conducted empirical studies with fixed target computational budgets despite its importance to resource-bounded devices. We present Recursive Speculative Decoding (RSD), a novel tree-based method that samples draft tokens without replacement and maximizes the diversity of the tree. During RSD's drafting, the tree is built by either Gumbel-Top-$k$ trick that draws tokens without replacement in parallel or Stochastic Beam Search that samples sequences without replacement while early-truncating unlikely draft sequences and reducing the computational cost of LLM. We empirically evaluate RSD with Llama 2 and OPT models, showing that RSD outperforms the baseline methods, consistently for fixed draft sequence length and in most cases for fixed computational budgets at LLM., Comment: 82 pages, 9 figures, 54 tables

Machine Learning models increasingly face data integrity challenges due to the use of large-scale training datasets drawn from the Internet. We study what model developers can do if they detect that some data was manipulated or incorrect. Such manipulated data can cause adverse effects including vulnerability to backdoored samples, systemic biases, and reduced accuracy on certain input domains. Realistically, all manipulated training samples cannot be identified, and only a small, representative subset of the affected data can be flagged. We formalize Corrective Machine Unlearning as the problem of mitigating the impact of data affected by unknown manipulations on a trained model, only having identified a subset of the corrupted data. We demonstrate that the problem of corrective unlearning has significantly different requirements from traditional privacy-oriented unlearning. We find most existing unlearning methods, including retraining-from-scratch without the deletion set, require most of the manipulated data to be identified for effective corrective unlearning. However, one approach, Selective Synaptic Dampening, achieves limited success, unlearning adverse effects with just a small portion of the manipulated samples in our setting, which shows encouraging signs for future progress. We hope our work spurs research towards developing better methods for corrective unlearning and offers practitioners a new strategy to handle data integrity challenges arising from web-scale training. Code is available at https://github.com/drimpossible/corrective-unlearning-bench., Comment: Published in Transactions of Machine Learning Research (TMLR), 17 pages, 7 figures

Large language models have the ability to generate text that mimics patterns in their inputs. We introduce a simple Markov Chain sequence modeling task in order to study how this in-context learning (ICL) capability emerges. In our setting, each example is sampled from a Markov chain drawn from a prior distribution over Markov chains. Transformers trained on this task form \emph{statistical induction heads} which compute accurate next-token probabilities given the bigram statistics of the context. During the course of training, models pass through multiple phases: after an initial stage in which predictions are uniform, they learn to sub-optimally predict using in-context single-token statistics (unigrams); then, there is a rapid phase transition to the correct in-context bigram solution. We conduct an empirical and theoretical investigation of this multi-phase process, showing how successful learning results from the interaction between the transformer's layers, and uncovering evidence that the presence of the simpler unigram solution may delay formation of the final bigram solution. We examine how learning is affected by varying the prior distribution over Markov chains, and consider the generalization of our in-context learning of Markov chains (ICL-MC) task to $n$-grams for $n > 2$.

Recent advancements in language technology and Artificial Intelligence have resulted in numerous Language Models being proposed to perform various tasks in the legal domain ranging from predicting judgments to generating summaries. Despite their immense potential, these models have been proven to learn and exhibit societal biases and make unfair predictions. In this study, we explore the ability of Large Language Models (LLMs) to perform legal tasks in the Indian landscape when social factors are involved. We present a novel metric, $\beta$-weighted $\textit{Legal Safety Score ($LSS_{\beta}$)}$, which encapsulates both the fairness and accuracy aspects of the LLM. We assess LLMs' safety by considering its performance in the $\textit{Binary Statutory Reasoning}$ task and its fairness exhibition with respect to various axes of disparities in the Indian society. Task performance and fairness scores of LLaMA and LLaMA--2 models indicate that the proposed $LSS_{\beta}$ metric can effectively determine the readiness of a model for safe usage in the legal sector. We also propose finetuning pipelines, utilising specialised legal datasets, as a potential method to mitigate bias and improve model safety. The finetuning procedures on LLaMA and LLaMA--2 models increase the $LSS_{\beta}$, improving their usability in the Indian legal domain. Our code is publicly released.

Incident management for large cloud services is a complex and tedious process and requires significant amount of manual efforts from on-call engineers (OCEs). OCEs typically leverage data from different stages of the software development lifecycle [SDLC] (e.g., codes, configuration, monitor data, service properties, service dependencies, trouble-shooting documents, etc.) to generate insights for detection, root causing and mitigating of incidents. Recent advancements in large language models [LLMs] (e.g., ChatGPT, GPT-4, Gemini) created opportunities to automatically generate contextual recommendations to the OCEs assisting them to quickly identify and mitigate critical issues. However, existing research typically takes a silo-ed view for solving a certain task in incident management by leveraging data from a single stage of SDLC. In this paper, we demonstrate that augmenting additional contextual data from different stages of SDLC improves the performance of two critically important and practically challenging tasks: (1) automatically generating root cause recommendations for dependency failure related incidents, and (2) identifying ontology of service monitors used for automatically detecting incidents. By leveraging 353 incident and 260 monitor dataset from Microsoft, we demonstrate that augmenting contextual information from different stages of the SDLC improves the performance over State-of-The-Art methods.

Deep neural networks, including Convolutional Neural Networks (CNNs) and Visual Transformers (ViT), have achieved stunning success in medical image domain. We study thorax disease classification in this paper. Effective extraction of features for the disease areas is crucial for disease classification on radiographic images. While various neural architectures and training techniques, such as self-supervised learning with contrastive/restorative learning, have been employed for disease classification on radiographic images, there are no principled methods which can effectively reduce the adverse effect of noise and background, or non-disease areas, on the radiographic images for disease classification. To address this challenge, we propose a novel Low-Rank Feature Learning (LRFL) method in this paper, which is universally applicable to the training of all neural networks. The LRFL method is both empirically motivated by the low frequency property observed on all the medical datasets in this paper, and theoretically motivated by our sharp generalization bound for neural networks with low-rank features. In the empirical study, using a neural network such as a ViT or a CNN pre-trained on unlabeled chest X-rays by Masked Autoencoders (MAE), our novel LRFL method is applied on the pre-trained neural network and demonstrate better classification results in terms of both multiclass area under the receiver operating curve (mAUC) and classification accuracy.

The advent of Large Language Models (LLMs) is promising and has found application in numerous fields, but as it often is with the medical field, the bar is typically quite high [5]. In tandem with LLMs, vector embedding models and vector databases provide a robust way of expressing numerous modes of data that are easily digestible by typical machine learning models. Along with the ease of adding information, knowledge, and data to these vector databases, they provide a compelling reason to apply them in numerous fields where the task of retrieving information is typically done by humans. Researchers at Google have developed a clear alternative model, Med-PaLM [6] specifically designed to match a clinician's level of accuracy when it comes to medical knowledge. When training classifiers, and developing models, it is imperative to maintain factuality and reduce bias [4]. Here, we explore the use of vector databases and embedding models as a means of encoding, and classifying text with the example and application in the field of medicine. We show the robustness of these tools depends heavily on the sparsity of the data presented, and even with low amounts of data in the vector database itself, the vector database does a good job at classifying data [9]. Using various LLMs to generate the medical data, we also understand the limitations of the medical knowledge of these models and encourage further expert medical review of our testing data. By using vector databases to classify a clinician's notes on a patient presented with a certain ailment, we understand the limitations of such methods, but also the promise of their prospective use and with continued testing and experimentation, hope to explore a unique use case of vector databases and embedding models., Comment: 11 pages, 8 figures, All robustness tests are in a linked pdf

Traditional Radiance Field (RF) representations capture details of a specific scene and must be trained afresh on each scene. Semantic feature fields have been added to RFs to facilitate several segmentation tasks. Generalised RF representations learn the principles of view interpolation. A generalised RF can render new views of an unknown and untrained scene, given a few views. We present a way to distil feature fields into the generalised GNT representation. Our GSN representation generates new views of unseen scenes on the fly along with consistent, per-pixel semantic features. This enables multi-view segmentation of arbitrary new scenes. We show different semantic features being distilled into generalised RFs. Our multi-view segmentation results are on par with methods that use traditional RFs. GSN closes the gap between standard and generalisable RF methods significantly. Project Page: https://vinayak-vg.github.io/GSN/, Comment: Accepted at the Main Technical Track of AAAI 2024

The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type A. As a first step to a classification, we analyse $A_n$-invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.

Extremum seeking control (ESC) constitutes a powerful technique for online optimization with theoretical guarantees for convergence to the neighborhood of the optimizer under well-understood conditions. However, ESC requires a nonconstant perturbation signal to provide persistent excitation to the target system to yield convergent results, which usually results in steady state oscillations. While certain techniques have been proposed to eliminate perturbations once the neighborhood of the minimizer is reached, system modifications and environmental perturbations can suddenly change the minimizer and nonconstant perturbations would once more be required to convergence to the new minimizer. Hence, this paper develops a retrospective cost-based ESC(RC/ESC) technique for online output minimization with a vanishing perturbation, that is, a perturbation that becomes zero as time increases independently from the state of the controller or the controlled system. The performance of the proposed algorithm is illustrated via numerical examples.

This paper presents a model-based, adaptive, nonlinear controller for the bicopter stabilization and trajectory-tracking problem. The nonlinear controller is designed using the backstepping technique. Due to the non-invertibility of the input map, the bicopter system is first dynamically extended. However, the resulting dynamically extended system is in the pure feedback form with the uncertainty appearing in the input map. The adaptive backstepping technique is then extended and applied to design the controller. The proposed controller is validated in simulation for a smooth and nonsmooth trajectory-tracking problem., Comment: arXiv admin note: text overlap with arXiv:2305.03554

Augmenting large language models (LLMs) to understand audio -- including non-speech sounds and non-verbal speech -- is critically important for diverse real-world applications of LLMs. In this paper, we propose Audio Flamingo, a novel audio language model with 1) strong audio understanding abilities, 2) the ability to quickly adapt to unseen tasks via in-context learning and retrieval, and 3) strong multi-turn dialogue abilities. We introduce a series of training techniques, architecture design, and data strategies to enhance our model with these abilities. Extensive evaluations across various audio understanding tasks confirm the efficacy of our method, setting new state-of-the-art benchmarks. Our demo website is https://audioflamingo.github.io/ and the code is open-sourced at https://github.com/NVIDIA/audio-flamingo., Comment: ICML 2024

This paper describes continuous-space methodologies to estimate the collision probability, Euclidean distance and gradient between an ellipsoidal robot model and an environment surface modeled as a set of Gaussian distributions. Continuous-space collision probability estimation is critical for uncertainty-aware motion planning. Most collision detection and avoidance approaches assume the robot is modeled as a sphere, but ellipsoidal representations provide tighter approximations and enable navigation in cluttered and narrow spaces. State-of-the-art methods derive the Euclidean distance and gradient by processing raw point clouds, which is computationally expensive for large workspaces. Recent advances in Gaussian surface modeling (e.g. mixture models, splatting) enable compressed and high-fidelity surface representations. Few methods exist to estimate continuous-space occupancy from such models. They require Gaussians to model free space and are unable to estimate the collision probability, Euclidean distance and gradient for an ellipsoidal robot. The proposed methods bridge this gap by extending prior work in ellipsoid-to-ellipsoid Euclidean distance and collision probability estimation to Gaussian surface models. A geometric blending approach is also proposed to improve collision probability estimation. The approaches are evaluated with numerical 2D and 3D experiments using real-world point cloud data. Methods for efficient calculation of these quantities are demonstrated to execute within a few microseconds per ellipsoid pair using a single-thread on low-power CPUs of modern embedded computers

This paper uses topological data analysis (TDA) tools and introduces a data-driven clustering-based stock selection strategy tailored for sparse portfolio construction. Our asset selection strategy exploits the topological features of stock price movements to select a subset of topologically similar (different) assets for a sparse index tracking (Markowitz) portfolio. We introduce new distance measures, which serve as an input to the clustering algorithm, on the space of persistence diagrams and landscapes that consider the time component of a time series. We conduct an empirical analysis on the S\&P index from 2009 to 2020, including a study on the COVID-19 data to validate the robustness of our methodology. Our strategy to integrate TDA with the clustering algorithm significantly enhanced the performance of sparse portfolios across various performance measures in diverse market scenarios.

The problem of optimally measuring an analytic function of unknown local parameters each linearly coupled to a qubit sensor is well understood, with applications ranging from field interpolation to noise characterization. Here, we resolve a number of open questions that arise when extending this framework to Mach-Zehnder interferometers and quadrature displacement sensing. In particular, we derive lower bounds on the achievable mean square error in estimating a linear function of either local phase shifts or quadrature displacements. In the case of local phase shifts, these results prove, and somewhat generalize, a conjecture by Proctor et al. [arXiv:1702.04271 (2017)]. For quadrature displacements, we extend proofs of lower bounds to the case of arbitrary linear functions. We provide optimal protocols achieving these bounds up to small (multiplicative) constants and describe an algebraic approach to deriving new optimal protocols, possibly subject to additional constraints. Using this approach, we prove necessary conditions for the amount of entanglement needed for any optimal protocol for both local phase and displacement sensing., Comment: 20 pages

Topic models are a popular tool for understanding text collections, but their evaluation has been a point of contention. Automated evaluation metrics such as coherence are often used, however, their validity has been questioned for neural topic models (NTMs) and can overlook a models benefits in real world applications. To this end, we conduct the first evaluation of neural, supervised and classical topic models in an interactive task based setting. We combine topic models with a classifier and test their ability to help humans conduct content analysis and document annotation. From simulated, real user and expert pilot studies, the Contextual Neural Topic Model does the best on cluster evaluation metrics and human evaluations; however, LDA is competitive with two other NTMs under our simulated experiment and user study results, contrary to what coherence scores suggest. We show that current automated metrics do not provide a complete picture of topic modeling capabilities, but the right choice of NTMs can be better than classical models on practical task., Comment: 19 pages, 5 tables, 6 figures, Accepted to EACL Main Conference 2024

Attitude filtering is a critical technology with applications in diverse domains such as aerospace engineering, robotics, computer vision, and augmented reality. Although attitude filtering is a particular case of the state estimation problem, attitude filtering is uniquely challenging due to the special geometric structure of the attitude parameterization. This paper presents a novel data-driven attitude filter, called the retrospective cost attitude filter (RCAF), for the SO(3) attitude representation. Like the multiplicative extended Kalman filter, RCAF uses a multiplicative correction signal, but instead of computing correction gains using Jacobians, RCAF computes the corrective signal using retrospective cost optimization and measured data. The RCAF filter is validated numerically in a scenario with noisy attitude measurements and noisy and biased rate-gyro measurements.

Participatory budgeting is a popular method to engage residents in budgeting decisions by local governments. The Stanford Participatory Budgeting platform is an online platform that has been used to engage residents in more than 150 budgeting processes. We present a data set with anonymized budget opinions from these processes with K-approval, K-ranking or knapsack primary ballots. For a subset of the voters, it includes paired votes with a different elicitation method in the same process. This presents a unique data set, as the voters, projects and setting are all related to real-world decisions that the voters have an actual interest in. With data from primary ballots we find that while ballot complexity (number of projects to choose from, number of projects to select and ballot length) is correlated with a higher median time spent by voters, it is not correlated with a higher abandonment rate. We use vote pairs with different voting methods to analyze the effect of voting methods on the cost of selected projects, more comprehensively than was previously possible. In most elections, voters selected significantly more expensive projects using K-approval than using knapsack, although we also find a small number of examples with a significant effect in the opposite direction. This effect happens at the aggregate level as well as for individual voters, and is influenced both by the implicit constraints of the voting method and the explicit constraints of the voting interface. Finally, we validate the use of K-ranking elicitation to offer a paper alternative for knapsack voting., Comment: Accepted for publication at ICWSM. Data set is available through: https://doi.org/10.25740/db709zg9088

The popular qubit framework has dominated recent work on quantum kernel machine learning, with results characterising expressivity, learnability and generalisation. As yet, there is no comparative framework to understand these concepts for continuous variable (CV) quantum computing platforms. In this paper we represent CV quantum kernels as closed form functions and use this representation to provide several important theoretical insights. We derive a general closed form solution for all CV quantum kernels and show every such kernel can be expressed as the product of a Gaussian and an algebraic function of the parameters of the feature map. Furthermore, in the multi-mode case, we present quantification of a quantum-classical separation for all quantum kernels via a hierarchical notion of the "stellar rank" of the quantum kernel feature map. We then prove kernels defined by feature maps of infinite stellar rank, such as GKP-state encodings, can be approximated arbitrarily well by kernels defined by feature maps of finite stellar rank. Finally, we simulate learning with a single-mode displaced Fock state encoding and show that (i) accuracy on our specific task (an annular data set) increases with stellar rank, (ii) for underfit models, accuracy can be improved by increasing a bandwidth hyperparameter, and (iii) for noisy data that is overfit, decreasing the bandwidth will improve generalisation but does so at the cost of effective stellar rank., Comment: 23+26 pages, 7 figures

Modular AI systems can be developed using LLM-prompts-based modules to minimize deployment time even for complex tasks. However, these systems do not always perform well and improving them using the data traces collected from a deployment remains an open challenge. The data traces contain LLM inputs and outputs, but the annotations from LLMs are noisy. We hypothesize that Active Label Correction (ALC) can be use on the collected data to train smaller task-specific improved models that can replace LLM-based modules. In this paper, we study the noise in three GPT-3.5-annotated datasets and their denoising with human feedback. We also propose a novel method ALC3 that iteratively applies three updates to the training dataset: auto-correction, correction using human feedback and filtering. Our results show that ALC3 can lead to oracle performance with feedback on 17-24% fewer examples than the number of noisy examples in the dataset across three different NLP tasks., Comment: EMNLP (Main) 2024, 13 pages, 6 figures

Assuming the generalized Lindel\"of hypothesis, we provide asymptotic formulas for the mean values of the first and second moments of Ramanujan sums over any number field. Additionally, unconditionally, we estimate the second moment of Ramanujan sums over cyclotomic number fields., Comment: 15 pages. arXiv admin note: text overlap with arXiv:2109.09398

For $n,d\in\mathbb{N}$, the cone $\mathcal{P}_{n+1,2d}$ of positive semi-definite (PSD) $(n+1)$-ary $2d$-ic forms (i.e., homogeneous polynomials with real coefficients in $n+1$ variables of degree $2d$) contains the cone $\Sigma_{n+1,2d}$ of those that are representable as finite sums of squares (SOS) of $(n+1)$-ary $d$-ic forms. Hilbert's 1888 Theorem states that $\Sigma_{n+1,2d}=\mathcal{P}_{n+1,2d}$ exactly in the Hilbert cases $(n+1,2d)$ with $n+1=2$ or $2d=2$ or $(3,4)$. For the non-Hilbert cases, we examine in [GHK] a specific cone filtration \begin{equation} \Sigma_{n+1,2d}=C_0\subseteq \ldots \subseteq C_n \subseteq C_{n+1} \subseteq \ldots \subseteq C_{k(n,d)-n}=\mathcal{P}_{n+1,2d}\end{equation} along $k(n,d)+1-n$ projective varieties containing the Veronese variety via the Gram matrix method. Here, $k(n,d)+1$ is the dimension of the real vector space of $(n+1)$-ary $d$-ic forms. In particular, we compute the number $\mu(n,d)$ of strictly separating intermediate cones (i.e., $C_i$ such that $\Sigma_{n+1,2d}\subsetneq C_i \subsetneq \mathcal{P}_{n+1,2d}$) for the cases $(3,6)$ and $(n+1,2d)_{n\geq 3,d=2,3}$. In this paper, firstly, we generalize our findings from [GHK] to any non-Hilbert case by identifying each strict inclusion in the above cone filtration. This allows us to give a refinement of Hilbert's 1888 Theorem by computing $\mu(n,d)$. The above cone filtration thus reduces to a specific cone subfiltration \begin{equation} \Sigma_{n+1,2d}=C_0^\prime\subsetneq C_1^\prime \subsetneq \ldots \subsetneq C_{\mu(n,d)}^\prime \subsetneq C_{\mu(n,d)+1}^\prime=\mathcal{P}_{n+1,2d} \end{equation} in which each inclusion is strict. Secondly, we show that each $C_i^\prime$, and hence each strictly separating $C_i$, fails to be a spectrahedral shadow., Comment: [GHK]: arXiv:2303.13178