Optimal approximate matrix product in terms of stable rank

We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having $m = O(\tilde{r}/\varepsilon^2)$ rows. Here $\tilde{r}$ is the maximum stable rank, i.e. squared ratio of Frobenius and operator norms, of the two matrices being multiplied. This is a quantitative improvement over previous work of [MZ11, KVZ14], and is also optimal for any oblivious dimensionality-reducing map. Furthermore, due to the black box reliance on the subspace embedding property in our proofs, our theorem can be applied to a much more general class of sketching matrices than what was known before, in addition to achieving better bounds. For example, one can apply our theorem to efficient subspace embeddings such as the Subsampled Randomized Hadamard Transform or sparse subspace embeddings, or even with subspace embedding constructions that may be developed in the future. Our main theorem, via connections with spectral error matrix multiplication shown in prior work, implies quantitative improvements for approximate least squares regression and low rank approximation. Our main result has also already been applied to improve dimensionality reduction guarantees for $k$-means clustering [CEMMP14], and implies new results for nonparametric regression [YPW15]. We also separately point out that the proof of the "BSS" deterministic row-sampling result of [BSS12] can be modified to show that for any matrices $A, B$ of stable rank at most $\tilde{r}$, one can achieve the spectral norm guarantee for approximate matrix multiplication of $A^T B$ by deterministically sampling $O(\tilde{r}/\varepsilon^2)$ rows that can be found in polynomial time. The original result of [BSS12] was for rank instead of stable rank. Our observation leads to a stronger version of a main theorem of [KMST10].Comment: v3: minor edits; v2: fixed one step in proof of Theorem 9 which was wrong by a constant factor (see the new Lemma 5 and its use; final theorem unaffected

A statistical analysis of the intercuspal angle of the phragmodontiform element of Phragmodus undatus

The intercuspal angle of the phragmodontiform element of the conodont Phragmodus undatus Branson and Mehl, 1933 was measured on specimens from samples from two cores drilled in the Cincinnati Region. In the section studied from the core from Minerva, Kentucky, a general trend can be observed that suggests a decrease in the mean intercuspal angle of the population with time. That trend correlates to a similar decrease in the relative abundance of P. undatus, which suggests a general shallowing of the water column. The section studied from the core from New Point, Indiana correlates to the upper part of the section studied from the core from Minerva, Kentucky, and extends stratigraphically higher. That section represents relatively deeper water, in which the mean intercuspal angle of the P. undatus population does not significantly differ with time. This suggests that a period of stasis developed during which P. undatus was in general equilibrium with paleoenvironmental factors as controlled by relative water depth.No embarg

Bacteria and the cellular basis of consciousness

According to Reber’s theory, the Cellular Basis of Consciousness (CBC), sentience originates as bio-sensitivity in unicellular organisms. For this reason, Reber regards sentience as evolutionarily foundational. Many bacteria show chemotaxis and, thus, according to CBC, they are sentient. Analysis of the genetic mechanisms underlying bacterial chemotaxis indicates that sentience has no explanatory power in this case. Genetic analysis also fails to show species continuity underlying bio-sensitivity in bacteria and bio-sensitivity in species with nervous systems, so it does not seem that sentience is evolutionary foundational. CBC is rejected on these grounds

The fish in the creek is sentient, even if I can’t speak with it

In this paper I argue that Velmens’ reflexive model of perceptual consciousness is useful for understanding the first-person perspective and sentience in animals. I then offer a defense of the proposal that ray-finned bony fish have a first-person perspective and sentience. This defense has two prongs. The first prong is presence of a substantial body of evidence that the neuroanatomy of the fish brain exhibits basic organizational principles associated with consciousness in mammals. These principles include a relationship between a second-order sensory relay, the preglomerular complex, and the fish pallium which bears a resemblance to the relationship between the mammalian thalamus and the neocortex, the existence of feedback/feedforward and reentrant circuitry in the pallium, and structural and functional differences among divisions of the fish pallium. The second prong is the existence of behaviors in fish that exhibit significant flexibility in the presence of environmental change and require relational learning among stimuli distributed in space, over time, or both. I conclude that, although they are instantiated differently, a first-person perspective and sentience are present in fish

Pain in fish: Evidence from peripheral nociceptors to pallial processing

The target article by Sneddon et al. (2018) presents convincing behavioral and pharmacological evidence that ray-finned fish consciously perceive noxious stimuli as painful. One objection to this interpretation of the evidence is that the fish nervous system is not complex enough to support the conscious experience of pain. Data that contradict this objection are presented in this commentary. The neuroanatomy and neurophysiology of the fish nervous system from the peripheral nerves to the pallium is able to support the sentient appreciation of pain

Sentience in fishes: More on the evidence

In my target article, I argued that the brains of ray-finned fishes of the teleost subclass (Actinopterygii) are sufficiently complex to support sentience — that these fishes have subjective awareness of interoceptive and exteroceptive sense experience. Extending previous theories centered on the tectum, I focused on the organization of the fish pallium. In this Response to the commentaries, I clarify that I do not propose that the fish pallium is, or must be, homologous to the mammalian neocortex to play a role in sentience. Some form of a functionalist approach to explaining the neural basis of sentience across taxa is probably most appropriate. However, what is known about the neural correlates of consciousness in humans is adequate to provide a starting place for investigation of the correlates of sentience in other animals, including fishes. Ultimately, though, hypotheses and experiments to evaluate anatomical and physiological correlates specific to sentience in fishes will be necessary

Scientific uncertainty and the animal sentience precautionary principle

Jonathan Birch offers the animal sentience precautionary principle (ASPP) as a framework for assigning sentience to animals. In doing this, he defines a BAR which when crossed will lead to action (ACT) and implementation of the ASPP. His effort to create a clear empirical basis for implementation of the precautionary principle in the area of animal welfare regulation is important. I argue, however, that his BAR is so low that the evidence supporting ACT is in danger of being overwhelmed by the problems of induction and the underdetermination of theory by evidence. If this happens, policy makers might well disregard the ASPP and fail to include sentient species in animal welfare regulation. I suggest that the BAR needs to be raised by inclusion of more independent indicators of sentience than those required by Birch

Implicit mental processes are an improbable basis for personhood

Rowlands argues that animals have implicit pre-reflective awareness and that this is adequate to create the unity of conscious thought required for personhood. For him pre-reflective awareness does not include intentionality and is probably an unconscious process. I suggest that his sense of implicit leads to significant difficulties for his argument and that including intentionality in the definition of a first-person perspective provides a stronger base for viewing animals as persons

Whether invertebrates are sentient matters to bioethics and science policy

Mikhalevich & Powell provide convincing empirical evidence that at least some invertebrates are sentient and hence should be granted moral status. I agree and argue that functional markers should be the primary indicators of sentience. Neuroanatomical homologies provide only secondary evidence. Consensus regarding the validity of these functional markers will be difficult to achieve. To be effective in practice, functional markers of sentience will have to be tested and accepted species by species to overcome the implicit biases against extending moral status to invertebrates